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    Wyświetlanie 1-35 z 35
    Tytuł:
    A Note on Total Graphs
    Autorzy:
    Forouhandeh, S.F.
    Jafari Rad, N.
    Vaqari Motlagh, B.H.
    Patil, H.P.
    Pandiya Raj, R.
    Powiązania:
    https://bibliotekanauki.pl/articles/31339326.pdf
    Data publikacji:
    2015-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total graph
    central graph
    middle graph
    Mycielski graph
    Opis:
    Erratum Identification and corrections of the existing mistakes in the paper On the total graph of Mycielski graphs, central graphs and their covering numbers, Discuss. Math. Graph Theory 33 (2013) 361-371.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2015, 35, 3; 585-587
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Radio Number of Cycles and their Total Graphs
    Autorzy:
    Merlin, E. T.
    Mangam, Tabitha Agnes
    Powiązania:
    https://bibliotekanauki.pl/articles/1177700.pdf
    Data publikacji:
    2018
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Radio labeling
    Radio number
    Total graph
    Opis:
    A radio labeling f of G is an assignment of positive integers to the vertices of G satisfying, │f (u) – f (v)│≥ diam(G) + 1 - d (u ,v) ∀ u, v ∈ V (G) where d (u ,v) is the distance between any two vertices in the graph. The radio number denoted by rn (G) is the minimum span of a radio labeling for G. In this paper, an alternate proof for radio number of cycles and exact radio number for their total graphs has been discussed.
    Źródło:
    World Scientific News; 2018, 101; 55-64
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Intersection graph of gamma sets in the total graph
    Autorzy:
    Chelvam, T.
    Asir, T.
    Powiązania:
    https://bibliotekanauki.pl/articles/743214.pdf
    Data publikacji:
    2012
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total graph
    gamma sets
    intersection graph
    Hamiltonian
    coloring
    connectivity
    domination number
    Opis:
    In this paper, we consider the intersection graph $I_{Γ}(ℤₙ)$ of gamma sets in the total graph on ℤₙ. We characterize the values of n for which $I_{Γ}(ℤₙ)$ is complete, bipartite, cycle, chordal and planar. Further, we prove that $I_{Γ}(ℤₙ)$ is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of $I_{Γ}(ℤₙ)$.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2012, 32, 2; 341-356
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Erdős-Gallai-Type Results for Total Monochromatic Connection of Graphs
    Autorzy:
    Jiang, Hui
    Li, Xueliang
    Zhang, Yingying
    Powiązania:
    https://bibliotekanauki.pl/articles/31343240.pdf
    Data publikacji:
    2019-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total-colored graph
    total monochromatic connection
    Erdős- Gallai-type problem
    Opis:
    A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a total monochromatically-connecting coloring (TMC-coloring, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color. For a connected graph G, the total monochromatic connection number, denoted by tmc(G), is defined as the maximum number of colors used in a TMC-coloring of G. In this paper, we study two kinds of Erdős-Gallai-type problems for tmc(G) and completely solve them.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2019, 39, 4; 775-785
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On the Total Graph of Mycielski Graphs, Central Graphs and Their Covering Numbers
    Autorzy:
    Patil, H.P.
    Pandiya Raj, R.
    Powiązania:
    https://bibliotekanauki.pl/articles/30146583.pdf
    Data publikacji:
    2013-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total graph
    central graph
    middle graph
    Mycielski graph
    independence number
    covering number
    edge independence number
    edge covering number
    chromatic number
    achromatic number
    Opis:
    The technique of counting cliques in networks is a natural problem. In this paper, we develop certain results on counting of triangles for the total graph of the Mycielski graph or central graph of star as well as completegraph families. Moreover, we discuss the upper bounds for the number of triangles in the Mycielski and other well known transformations of graphs. Finally, it is shown that the achromatic number and edge-covering number of the transformations mentioned above are equated.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2013, 33, 2; 361-371
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Grundy Total Domination Number in Product Graphs
    Autorzy:
    Brešar, Boštjan
    Bujtás, Csilla
    Gologranc, Tanja
    Klavžar, Sandi
    Košmrlj, Gašper
    Marc, Tilen
    Patkós, Balázs
    Tuza, Zsolt
    Vizer, Máté
    Powiązania:
    https://bibliotekanauki.pl/articles/32083828.pdf
    Data publikacji:
    2021-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total domination
    Grundy total domination number
    graph product
    Opis:
    A longest sequence $(v_1, . . ., v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, \(N(υ_i)\backslash\bigcup_{j=1}^{i-1}N(υ_j)≠∅\). The length $k$ of the sequence is called the Grundy total domination number of $G$ and denoted $\gamma_{gr}^t(G)$. In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that $\gamma_{gr}^t(G×H)≥\gamma_{gr}^t(G)\gamma_{gr}^t(H)$, conjecture that the equality always holds, and prove the conjecture in several special cases. For the lexicographic product we express $\gamma_{gr}^t(G∘H)$ in terms of related invariant of the factors and find some explicit formulas for it. For the strong product, lower bounds on $\gamma_{gr}^t(G⊠H)$ are proved as well as upper bounds for products of paths and cycles. For the Cartesian product we prove lower and upper bounds on the Grundy total domination number when factors are paths or cycles.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 1; 225-247
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Total Colorings of Embedded Graphs with No 3-Cycles Adjacent to 4-Cycles
    Autorzy:
    Wang, Bing
    Wu, Jian-Liang
    Sun, Lin
    Powiązania:
    https://bibliotekanauki.pl/articles/31342246.pdf
    Data publikacji:
    2018-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total coloring
    embedded graph
    cycle
    Opis:
    A total-k-coloring of a graph G is a coloring of V ∪ E using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ′′(G) of G is the smallest integer k such that G has a total-k-coloring. Let G be a graph embedded in a surface of Euler characteristic ε ≥ 0. If G contains no 3-cycles adjacent to 4-cycles, that is, no 3-cycle has a common edge with a 4-cycle, then χ′′(G) ≤ max{8, Δ+1}.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2018, 38, 4; 977-989
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Fair Total Domination Number in Cactus Graphs
    Autorzy:
    Hajian, Majid
    Rad, Nader Jafari
    Powiązania:
    https://bibliotekanauki.pl/articles/32083904.pdf
    Data publikacji:
    2021-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    fair total domination
    cactus graph
    Opis:
    For k ≥ 1, a k-fair total dominating set (or just kFTD-set) in a graph G is a total dominating set S such that |N(v) ∩ S| = k for every vertex v ∈ V\S. The k-fair total domination number of G, denoted by ftdk(G), is the minimum cardinality of a kFTD-set. A fair total dominating set, abbreviated FTD-set, is a kFTD-set for some integer k ≥ 1. The fair total domination number of a nonempty graph G, denoted by ftd(G), of G is the minimum cardinality of an FTD-set in G. In this paper, we present upper bounds for the 1-fair total domination number of cactus graphs, and characterize cactus graphs achieving equality for the upper bounds.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 2; 647-664
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Total Coloring of Claw-Free Planar Graphs
    Autorzy:
    Liang, Zuosong
    Powiązania:
    https://bibliotekanauki.pl/articles/32304195.pdf
    Data publikacji:
    2022-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total coloring
    total coloring conjecture
    planar graph
    claw
    Opis:
    A total coloring of a graph is an assignment of colors to both its vertices and edges so that adjacent or incident elements acquire distinct colors. Let Δ(G) be the maximum degree of G. Vizing conjectured that every graph has a total (Δ + 2)-coloring. This Total Coloring Conjecture remains open even for planar graphs, for which the only open case is Δ = 6. Claw-free planar graphs have Δ ≤ 6. In this paper, we prove that the Total Coloring Conjecture holds for claw-free planar graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 3; 771-777
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A note on total colorings of planar graphs without 4-cycles
    Autorzy:
    Wang, Ping
    Wu, Jian-Liang
    Powiązania:
    https://bibliotekanauki.pl/articles/744436.pdf
    Data publikacji:
    2004
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total coloring
    planar graph
    list coloring
    girth
    Opis:
    Let G be a 2-connected planar graph with maximum degree Δ such that G has no cycle of length from 4 to k, where k ≥ 4. Then the total chromatic number of G is Δ +1 if (Δ,k) ∈ {(7,4),(6,5),(5,7),(4,14)}.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2004, 24, 1; 125-135
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Some totally 4-choosable multigraphs
    Autorzy:
    Woodall, Douglas
    Powiązania:
    https://bibliotekanauki.pl/articles/743409.pdf
    Data publikacji:
    2007
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    maximum average degree
    planar graph
    total choosability
    list total colouring
    Opis:
    It is proved that if G is multigraph with maximum degree 3, and every submultigraph of G has average degree at most 2(1/2) and is different from one forbidden configuration C⁺₄ with average degree exactly 2(1/2), then G is totally 4-choosable; that is, if every element (vertex or edge) of G is assigned a list of 4 colours, then every element can be coloured with a colour from its own list in such a way that no two adjacent or incident elements are coloured with the same colour. This shows that the List-Total-Colouring Conjecture, that ch''(G) = χ''(G) for every multigraph G, is true for all multigraphs of this type. As a consequence, if G is a graph with maximum degree 3 and girth at least 10 that can be embedded in the plane, projective plane, torus or Klein bottle, then ch''(G) = χ''(G) = 4. Some further total choosability results are discussed for planar graphs with sufficiently large maximum degree and girth.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2007, 27, 3; 425-455
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On total vertex irregularity strength of graphs
    Autorzy:
    Muthu Guru Packiam, K.
    Kathiresan, Kumarappan
    Powiązania:
    https://bibliotekanauki.pl/articles/743655.pdf
    Data publikacji:
    2012
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    graph labeling
    irregularity strength
    total assignment
    vertex irregular total labeling
    Opis:
    Martin Bača et al. [2] introduced the problem of determining the total vertex irregularity strengths of graphs. In this paper we discuss how the addition of new edge affect the total vertex irregularity strength.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2012, 32, 1; 39-45
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Total edge irregularity strength of trees
    Autorzy:
    Ivančo, Jaroslav
    Jendrol', Stanislav
    Powiązania:
    https://bibliotekanauki.pl/articles/743612.pdf
    Data publikacji:
    2006
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    graph labelling
    tree
    irregularity strength
    total labellings
    total edge irregularity strength
    Opis:
    A total edge-irregular k-labelling ξ:V(G)∪ E(G) → {1,2,...,k} of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-labelling is called the total edge irregularity strength of G, tes(G). In this paper we prove that for every tree T of maximum degree Δ on p vertices
    tes(T) = max{⎡(p+1)/3⎤,⎡(Δ+1)/2⎤}.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2006, 26, 3; 449-456
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On (p, 1)-Total Labelling of Some 1-Planar Graphs
    Autorzy:
    Niu, Bei
    Zhang, Xin
    Powiązania:
    https://bibliotekanauki.pl/articles/32083894.pdf
    Data publikacji:
    2021-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    1-planar graph
    total coloring
    structural theorem
    (p, 1)-total labelling
    Opis:
    A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that the (p, 1)-total labelling number (p ≥ 2) of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ (G) ≥ 6p + 7 or Δ (G) ≥ 4p + 6 and G is triangle-free.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 2; 531-558
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The List Edge Coloring and List Total Coloring of Planar Graphs with Maximum Degree at Least 7
    Autorzy:
    Sun, Lin
    Wu, Jianliang
    Wang, Bing
    Liu, Bin
    Powiązania:
    https://bibliotekanauki.pl/articles/31348158.pdf
    Data publikacji:
    2020-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    planar graph
    list edge coloring
    list total coloring
    Opis:
    A graph $G$ is edge $k$-choosable (respectively, total $k$-choosable) if, whenever we are given a list $L(x)$ of colors with $|L(x)| = k$ for each $x ∈ E(G) (x ∈ E(G) ∪ V (G))$, we can choose a color from $L(x)$ for each element $x$ such that no two adjacent (or incident) elements receive the same color. The list edge chromatic index $χ_l^′(G)$ (respectively, the list total chromatic number $χ_l^{′′}(G))$ of $G$ is the smallest integer $k$ such that $G$ is edge (respectively, total) $k$-choosable. In this paper, we focus on a planar graph $G$, with maximum degree $Δ (G) ≥ 7$ and with some structural restrictions, satisfies $χ_l^′(G) = Δ (G)$ and $χ_l^{′′}(G) = Δ (G) + 1$.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 4; 1005-1024
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Twin Minus Total Domination Numbers In Directed Graphs
    Autorzy:
    Dehgardi, Nasrin
    Atapour, Maryam
    Powiązania:
    https://bibliotekanauki.pl/articles/31341587.pdf
    Data publikacji:
    2017-11-27
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    twin minus total dominating function
    twin minus total domination number
    directed graph
    Opis:
    Let $ D = (V,A) $ be a finite simple directed graph (shortly, digraph). A function $ f : V \rightarrow {−1, 0, 1} $ is called a twin minus total dominating function (TMTDF) if $ f(N^−(v)) \ge 1 $ and $ f(N^+(v)) \ge 1 $ for each vertex $ v \in V $. The twin minus total domination number of $D$ is $\gamma_{mt}^\ast (D) = \text{min} \{ w(f) | f $ is a TMTDF of $ D \} $. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for $ \gamma_{mt}^\ast (D) $ in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 4; 989-1004
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Rainbow Total-Coloring of Complementary Graphs and Erdős-Gallai Type Problem For The Rainbow Total-Connection Number
    Autorzy:
    Sun, Yuefang
    Jin, Zemin
    Tu, Jianhua
    Powiązania:
    https://bibliotekanauki.pl/articles/31342242.pdf
    Data publikacji:
    2018-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Rainbow total-coloring
    rainbow total-connection number
    complementary graph
    Erdős-Gallai type problem
    Opis:
    A total-colored graph $G$ is rainbow total-connected if any two vertices of $G$ are connected by a path whose edges and internal vertices have distinct colors. The rainbow total-connection number, denoted by $ rtc(G) $, of a graph $G$ is the minimum number of colors needed to make $G$ rainbow total-connected. In this paper, we prove that $ rtc(G) $ can be bounded by a constant 7 if the following three cases are excluded: $ diam( \overline{G} ) = 2 $, $ diam( \overline{G} ) = 3 $, $ \overline{G} $ contains exactly two connected components and one of them is a trivial graph. An example is given to show that this bound is best possible. We also study Erdős-Gallai type problem for the rainbow total-connection number, and compute the lower bounds and precise values for the function $ f(n, k) $, where $ f(n, k) $ is the minimum value satisfying the following property: if $ |E(G)| \ge f(n, k) $, then $ rtc(G) \le k $.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2018, 38, 4; 1023-1036
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Supermagic Graphs Having a Saturated Vertex
    Autorzy:
    Ivančo, Jaroslav
    Polláková, Tatiana
    Powiązania:
    https://bibliotekanauki.pl/articles/30147220.pdf
    Data publikacji:
    2014-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    supermagic graph
    saturated vertex
    vertex-magic total labeling
    Opis:
    A graph is called supermagic if it admits a labeling of the edges by pairwise different consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we establish some conditions for graphs with a saturated vertex to be supermagic. Inter alia we show that complete multipartite graphs K1,n,n and K1,2,...,2 are supermagic.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2014, 34, 1; 75-84
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On local antimagic total labeling of complete graphs amalgamation
    Autorzy:
    Lau, Gee-Choon
    Shiu, Wai-Chee
    Powiązania:
    https://bibliotekanauki.pl/articles/29519348.pdf
    Data publikacji:
    2023
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    local antimagic (total) chromatic number
    amalgamation
    complete graph
    Opis:
    Let G = (V,E) be a connected simple graph of order p and size q. A graph G is called local antimagic (total) if G admits a local antimagic (total) labeling. A bijection g : E → {1, 2, . . . , q} is called a local antimagic labeling of G if for any two adjacent vertices u and v, we have $ g^+ (u) \ne g^+ (v) $, where $ g^+ (u) = \Sigma_{e∈E(u)} \text{ } g(e) $, and E(u) is the set of edges incident to u. Similarly, a bijection f : V (G)∪E(G) → {1, 2, . . . , p+q} is called a local antimagic total labeling of G if for any two adjacent vertices u and v, we have $ w_f (u) \ne w_f (v) $, where $ w_f (u) = f(u) + \Sigma_{e∈E(u)} f(e) $. Thus, any local antimagic (total) labeling induces a proper vertex coloring of G if vertex v is assigned the color $g^+ (v) $ (respectively, $ w_f (u) $). The local antimagic (total) chromatic number, denoted $χ_\text{la } (G) $ (respectively $χ_\text{lat } (G) $ ), is the minimum number of induced colors taken over local antimagic (total) labeling of G. In this paper, we determined $ χ_\text{lat } (G) $ where G is the amalgamation of complete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of $ K_1 $ and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.
    Źródło:
    Opuscula Mathematica; 2023, 43, 3; 429-453
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Neighbor Product Distinguishing Total Colorings of Planar Graphs with Maximum Degree at least Ten
    Autorzy:
    Dong, Aijun
    Li, Tong
    Powiązania:
    https://bibliotekanauki.pl/articles/32227944.pdf
    Data publikacji:
    2021-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total coloring
    neighbor product distinguishing coloring
    planar graph
    Opis:
    A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2, . . ., k}. Let p(u) denote the product of the color on a vertex u and colors on all the edges incident with u. For each edge uv ∈ E(G), if p(u) ≠ p(v), then we say the coloring c distinguishes adjacent vertices by product and call it a neighbor product distinguishing k-total coloring of G. By X″∏(G), we denote the smallest value of k in such a coloring of G. It has been conjectured by Li et al. that Δ(G) + 3 colors enable the existence of a neighbor product distinguishing total coloring. In this paper, by applying the Combinatorial Nullstellensatz, we obtain that the conjecture holds for planar graph with Δ(G) ≥ 10. Moreover, for planar graph G with Δ(G) ≥ 11, it is neighbor product distinguishing (Δ(G) + 2)-total colorable, and the upper bound Δ(G) + 2 is tight.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 4; 981-999
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Partitioning a graph into a dominating set, a total dominating set, and something else
    Autorzy:
    Henning, Michael
    Löwenstein, Christian
    Rautenbach, Dieter
    Powiązania:
    https://bibliotekanauki.pl/articles/744065.pdf
    Data publikacji:
    2010
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    domination
    total domination
    domatic number
    vertex partition
    Petersen graph
    Opis:
    A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159-162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D∪T necessarily contains all vertices of the graph.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2010, 30, 4; 563-574
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Zig-zag facial total-coloring of plane graphs
    Autorzy:
    Czap, J.
    Jendrol, S.
    Voigt, M.
    Powiązania:
    https://bibliotekanauki.pl/articles/255827.pdf
    Data publikacji:
    2018
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    plane graph
    facial coloring total-coloring zig-zag coloring
    Opis:
    In this paper we introduce the concept of zig-zag facial total-coloring of plane graphs. We obtain lower and upper bounds for the minimum number of colors which is necessary for such a coloring. Moreover, we give several sharpness examples and formulate some open problems.
    Źródło:
    Opuscula Mathematica; 2018, 38, 6; 819-827
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles
    Autorzy:
    Zhao, Xue
    Xu, Chang-Qing
    Powiązania:
    https://bibliotekanauki.pl/articles/32083807.pdf
    Data publikacji:
    2020-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    neighbor sum distinguishing total coloring
    discharging method
    planar graph
    Opis:
    For a given graph $ G = (V (G), E(G)) $, a proper total coloring $ \phi : V (G) \cup E(G) $ $ \rightarrow {1, 2, . . ., k} $ is neighbor sum distinguishing if $ f(u) \ne f(v) $ for each edge $ uv \in E(G) $, where $ f(v) = \Sigma_{ uv \in E(G) } $ $ \phi (uv) + \phi (v) $, $ v \in V (G) $. The smallest integer $k$ in such a coloring of $G$ is the neighbor sum distinguishing total chromatic number, denoted by $ \chi_\Sigma^{''} (G) $. Pilśniak and Woźniak first introduced this coloring and conjectured that $ \chi_\Sigma^{''}(G) \le \Delta (G)+3 $ for any graph with maximum degree $ \Delta (G) $. In this paper, by using the discharging method, we prove that for any planar graph $G$ without 5-cycles, $ \chi_\Sigma^{''} (G) \le \text{max} \{ \Delta (G)+2, 10 \} $. The bound $ \Delta (G) + 2 $ is sharp. Furthermore, we get the exact value of $ \chi_\Sigma^{''} (G) $ if $ \Delta (G) \ge 9 $.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 1; 243-253
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On List Equitable Total Colorings of the Generalized Theta Graph
    Autorzy:
    Mudrock, Jeffrey A.
    Marsh, Max
    Wagstrom, Tim
    Powiązania:
    https://bibliotekanauki.pl/articles/32326107.pdf
    Data publikacji:
    2021-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    graph coloring
    total coloring
    equitable coloring
    list coloring
    equitable choosability
    Opis:
    In 2003, Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. A k-assignment, L, for a graph G assigns a list, L(v), of k available colors to each v ∈ V (G), and an equitable L-coloring of G is a proper coloring, f, of G such that f(v) ∈ L(v) for each v ∈ V (G) and each color class of f has size at most ⌈|V (G)|/k⌉. Graph G is equitably k-choosable if G is equitably L-colorable whenever L is a k-assignment for G. In 2018, Kaul, Mudrock, and Pelsmajer subsequently introduced the List Equitable Total Coloring Conjecture which states that if T is a total graph of some simple graph, then T is equitably k-choosable for each k ≥ max{xℓ(T), Δ(T)/2 + 2} where Δ(T) is the maximum degree of a vertex in T and xℓ(T ) is the list chromatic number of T. In this paper, we verify the List Equitable Total Coloring Conjecture for subdivisions of stars and the generalized theta graph.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 4; 1215-1233
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Weak Total Resolvability In Graphs
    Autorzy:
    Casel, Katrin
    Estrada-Moreno, Alejandro
    Fernau, Henning
    Rodríguez-Velázquez, Juan Alberto
    Powiązania:
    https://bibliotekanauki.pl/articles/31341108.pdf
    Data publikacji:
    2016-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    metric dimension
    resolving set
    weak total metric dimension
    weak total resolving set
    adjacency dimension
    graph operations
    Opis:
    A vertex v ∈ V (G) is said to distinguish two vertices x, y ∈ V (G) of a graph G if the distance from v to x is di erent from the distance from v to y. A set W ⊆ V (G) is a total resolving set for a graph G if for every pair of vertices x, y ∈ V (G), there exists some vertex w ∈ W − {x, y} which distinguishes x and y, while W is a weak total resolving set if for every x ∈ V (G)−W and y ∈ W, there exists some w ∈ W −{y} which distinguishes x and y. A weak total resolving set of minimum cardinality is called a weak total metric basis of G and its cardinality the weak total metric dimension of G. Our main contributions are the following ones: (a) Graphs with small and large weak total metric bases are characterised. (b) We explore the (tight) relation to independent 2-domination. (c) We introduce a new graph parameter, called weak total adjacency dimension and present results that are analogous to those presented for weak total dimension. (d) For trees, we derive a characterisation of the weak total (adjacency) metric dimension. Also, exact figures for our parameters are presented for (generalised) fans and wheels. (e) We show that for Cartesian product graphs, the weak total (adjacency) metric dimension is usually pretty small. (f) The weak total (adjacency) dimension is studied for lexicographic products of graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2016, 36, 1; 185-210
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Total connected domination game
    Autorzy:
    Bujtás, Csilla
    Henning, Michael A.
    Iršič, Vesna
    Klavžar, Sandi
    Powiązania:
    https://bibliotekanauki.pl/articles/2050904.pdf
    Data publikacji:
    2021
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    connected domination game
    total connected domination game
    graph product
    tree
    Opis:
    The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number $(\gamma_{tcg}(G))(\gamma_{cg(G)})$ of $G$. We show that $\gamma_{tcg}(G) \in \{\gamma_{cg}(G), \gamma_{cg}(G)+1, \gamma_{cg}(G)+2\}$, and consequently define $G$ as Class $i$ if $\gamma_{tcg}(G) = \gamma_{cg}(G)+i$ for $i \in \{0, 1, 2\}$. A large family of Class 0 graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minimum degree at least 2. We show that no tree is Class 2 and characterize Class 1 trees. We provide an infinite family of Class 2 bipartite graphs.
    Źródło:
    Opuscula Mathematica; 2021, 41, 4; 453-464
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Facial [r,s,t]-Colorings of Plane Graphs
    Autorzy:
    Czap, Július
    Šugerek, Peter
    Jendrol’, Stanislav
    Valiska, Juraj
    Powiązania:
    https://bibliotekanauki.pl/articles/31343366.pdf
    Data publikacji:
    2019-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    plane graph
    boundary walk
    edge-coloring
    vertex-coloring
    total-coloring
    Opis:
    Let $G$ be a plane graph. Two edges are facially adjacent in $G$ if they are consecutive edges on the boundary walk of a face of $G$. Given nonnegative integers $r$, $s$, and $t$, a facial $[r, s, t]$-coloring of a plane graph $G = (V,E)$ is a mapping $f : V \cup E \rightarrow {1, . . ., k} $ such that $ |f(v_1) − f(v_2)| \ge r $ for every two adjacent vertices $ v_1 $ and $ v_2 $, $ | f(e_1) − f(e_2)| \ge s $ for every two facially adjacent edges $ e_1 $ and $ e_2 $, and $ | f(v) − f(e)| \ge t $ for all pairs of incident vertices $ v $ and edges $ e $. The facial $[r, s, t]$-chromatic number $ \overline{ \chi }_{r,s,t} (G) $ of $ G $ is defined to be the minimum $k$ such that $G$ admits a facial $[r, s, t]$-coloring with colors $1, . . ., k$. In this paper we show that $ \overline{ \chi }_{r,s,t} (G) \le 3r + 3s + t + 1 $ for every plane graph $G$. For some triplets $ [r, s, t] $ and for some families of plane graphs this bound is improved. Special attention is devoted to the cases when the parameters $r$, $s$, and $t$ are small.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2019, 39, 3; 629-645
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Chromatic Properties of the Pancake Graphs
    Autorzy:
    Konstantinova, Elena
    Powiązania:
    https://bibliotekanauki.pl/articles/31341647.pdf
    Data publikacji:
    2017-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Pancake graph
    Cayley graphs
    symmetric group
    chromatic number
    total chromatic number
    Opis:
    Chromatic properties of the Pancake graphs Pn, n ⩾ 2, that are Cayley graphs on the symmetric group Symn generated by prefix-reversals are investigated in the paper. It is proved that for any n ⩾ 3 the total chromatic number of Pn is n, and it is shown that the chromatic index of Pn is n − 1. We present upper bounds on the chromatic number of the Pancake graphs Pn, which improve Brooks’ bound for n ⩾ 7 and Katlin’s bound for n ⩽ 28. Algorithms of a total n-coloring and a proper (n − 1)-coloring are given.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 3; 777-787
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The irregularity of graphs under graph operations
    Autorzy:
    Abdo, Hosam
    Dimitrov, Darko
    Powiązania:
    https://bibliotekanauki.pl/articles/30148232.pdf
    Data publikacji:
    2014-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    irregularity of graphs
    total irregularity of graphs
    graph operations
    Zagreb indices
    Opis:
    The irregularity of a simple undirected graph $G$ was defined by Albertson [5] as $irr(G) = ∑_{uv∈E(G)} |dG(u) − dG(v)|$, where $d_G(u)$ denotes the degree of a vertex $u ∈ V (G)$. In this paper we consider the irregularity of graphs under several graph operations including join, Cartesian product, direct product, strong product, corona product, lexicographic product, disjunction and symmetric difference. We give exact expressions or (sharp) upper bounds on the irregularity of graphs under the above mentioned operations
    Źródło:
    Discussiones Mathematicae Graph Theory; 2014, 34, 2; 263-278
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Edge colorings and total colorings of integer distance graphs
    Autorzy:
    Kemnitz, Arnfried
    Marangio, Massimiliano
    Powiązania:
    https://bibliotekanauki.pl/articles/743555.pdf
    Data publikacji:
    2002
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    integer distance graph
    chromatic number
    choice number
    chromatic index
    choice index
    total chromatic number
    total choice number
    Opis:
    An integer distance graph is a graph G(D) with the set Z of integers as vertex set and two vertices u,v ∈ Z are adjacent if and only if |u-v| ∈ D where the distance set D is a subset of the positive integers N. In this note we determine the chromatic index, the choice index, the total chromatic number and the total choice number of all integer distance graphs, and the choice number of special integer distance graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2002, 22, 1; 149-158
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A Note on the Total Detection Numbers of Cycles
    Autorzy:
    Escuadro, Henry E.
    Fujie, Futaba
    Musick, Chad E.
    Powiązania:
    https://bibliotekanauki.pl/articles/31339492.pdf
    Data publikacji:
    2015-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    vertex-distinguishing coloring
    detectable labeling
    detection number
    total detection number
    Hamiltonian graph
    Opis:
    Let G be a connected graph of size at least 2 and c :E(G)→{0, 1, . . ., k− 1} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . ., ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c is called a detectable labeling if distinct vertices in G have distinct color codes. The value val(c) of a detectable labeling c of a graph G is the sum of the labels assigned to the edges in G. The total detection number td(G) of G is defined by td(G) = min{val(c)}, where the minimum is taken over all detectable labelings c of G. We investigate the problem of determining the total detection numbers of cycles.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2015, 35, 2; 237-247
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs
    Autorzy:
    Song, Wen-Yao
    Miao, Lian-Ying
    Duan, Yuan-Yuan
    Powiązania:
    https://bibliotekanauki.pl/articles/32083736.pdf
    Data publikacji:
    2020-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    neighbor sum distinguishing total choosability
    maximum degree
    IC-planar graph
    Combinatorial Nullstellensatz
    Opis:
    Two distinct crossings are independent if the end-vertices of the crossed pair of edges are mutually different. If a graph $G$ has a drawing in the plane such that every two crossings are independent, then we call $G$ a plane graph with independent crossings or IC-planar graph for short. A proper total-$k$-coloring of a graph $G$ is a mapping $ c : V (G) \cup E(G) \rightarrow \{ 1, 2, . . ., k \} $ such that any two adjacent elements in $ V (G) \cup E(G) $ receive different colors. Let $ \Sigma_c (v) $ denote the sum of the color of a vertex $v$ and the colors of all incident edges of $v$. A total-$k$-neighbor sum distinguishing-coloring of $G$ is a total-$k$-coloring of $G$ such that for each edge $ uv \in E(G)$, $\Sigma_c (u) \ne \Sigma_c (v) $. The least number $k$ needed for such a coloring of $G$ is the neighbor sum distinguishing total chromatic number, denoted by $ \chi_\Sigma^{''} (G) $. In this paper, it is proved that if $G$ is an IC-planar graph with maximum degree $ \Delta (G) $, then $ ch_\Sigma^{''} (G) \le \text{max} \{ \Delta (G)+3, 17 \} $, where $ ch_\Sigma^{''} (G) $ is the neighbor sum distinguishing total choosability of $G$.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 1; 331-344
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Generalized Fractional and Circular Total Colorings of Graphs
    Autorzy:
    Kemnitz, Arnfried
    Marangio, Massimiliano
    Mihók, Peter
    Oravcová, Janka
    Soták, Roman
    Powiązania:
    https://bibliotekanauki.pl/articles/31339338.pdf
    Data publikacji:
    2015-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    graph property
    (P,Q)-total coloring
    circular coloring
    fractional coloring
    fractional (P,Q)-total chromatic number
    circular (P,Q)- total chromatic number
    Opis:
    Let \( \mathcal{P} \) and \( \mathcal{Q} \) be additive and hereditary graph properties, $ r, s \in \mathbb{N}$, $ r \ge s $, and $ [\mathbb{Z}_r]^s $ be the set of all s-element subsets of $\mathbb{Z}_r $. An ($r$, $s$)-fractional (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of $G$ is an assignment $ h : V (G) \cup E(G) \rightarrow [\mathbb{Z}_r]^s $ such that for each $ i \in \mathbb{Z}_r $ the following holds: the vertices of $G$ whose color sets contain color $i$ induce a subgraph of $G$ with property \( \mathcal{P} \), edges with color sets containing color $i$ induce a subgraph of $G$ with property \( \mathcal{Q} \), and the color sets of incident vertices and edges are disjoint. If each vertex and edge of $G$ is colored with a set of $s$ consecutive elements of $ \mathbb{Z}_r $ we obtain an ($r$, $s$)-circular (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of $G$. In this paper we present basic results on ($r$, $s$)-fractional/circular (\( \mathcal{P} \),\( \mathcal{Q} \))-total colorings. We introduce the fractional and circular (\( \mathcal{P} \),\( \mathcal{Q}\))-total chromatic number of a graph and we determine this number for complete graphs and some classes of additive and hereditary properties.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2015, 35, 3; 517-532
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Fractional (\( \mathcal{P} , \mathcal{Q} \))-Total List Colorings of Graphs
    Autorzy:
    Kemnitz, Arnfried
    Mihók, Peter
    Voigt, Margit
    Powiązania:
    https://bibliotekanauki.pl/articles/30146708.pdf
    Data publikacji:
    2013-03-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    graph property
    total coloring
    (P,Q)-total coloring
    fractional coloring
    fractional (P,Q)-total chromatic number
    circular coloring
    circular (P,Q)-total chromatic number
    list coloring
    (P,Q)-total (a
    b)-list colorings
    Opis:
    Let $ r, s \in \mathbb{N}$, $r \ge s$, and \( \mathcal{P} \) and \( \mathcal{Q} \) be two additive and hereditary graph properties. A \( (P,Q) \)-total $(r, s)$-coloring of a graph $G = (V,E)$ is a coloring of the vertices and edges of $G$ by $s$-element subsets of $ \mathbb{Z}_r$ such that for each color $i$, $0 \le i \le r − 1$, the vertices colored by subsets containing $i$ induce a subgraph of $G$ with property \( \mathcal{P} \), the edges colored by subsets containing $i$ induce a subgraph of $G$ with property \( \mathcal{Q} \), and color sets of incident vertices and edges are disjoint. The fractional \( (\mathcal{P}, \mathcal{Q})\)-total chromatic number $ \chi_{f,P,Q}^{''}(G)$ of $G$ is defined as the infimum of all ratios $r//s$ such that $G$ has a \( ( \mathcal{P}, \mathcal{Q})\)-total $(r, s)$-coloring. A \( ( \mathcal{P}, \mathcal{Q} \)-total independent set $ T = V_T \cup E_T \subseteq V \cup E$ is the union of a set $V_T$ of vertices and a set $E_T$ of edges of $G$ such that for the graphs induced by the sets $V_T$ and $E_T$ it holds that \( G[ V_T ] \in \mathcal{ P } \), \( G[ E_T ] \in \mathcal{Q} \), and $ G[ V_T ] $ and $ G[ E_T ] $ are disjoint. Let \( T_{ \mathcal{P} , \mathcal{Q} } \) be the set of all \( (\mathcal{P} ,\mathcal{Q})\)-total independent sets of $G$. Let $L(x)$ be a set of admissible colors for every element $ x \in V \cup E $. The graph $G$ is called \( (\mathcal{P} , \mathcal{Q}) \)-total $(a, b)$-list colorable if for each list assignment $L$ with $|L(x)| = a$ for all $x \in V \cup E$ it is possible to choose a subset $ C(x) \subseteq L(x)$ with $|C(x)| = b$ for all $ x \in V \cup E$ such that the set $ T_i $ which is defined by $ T_i = {x \in V \cup E : i \in C(x) } $ belongs to \( T_{ \mathcal{P},\mathcal{Q}}\) for every color $i$. The \( (\mathcal{P}, \mathcal{Q})\)- choice ratio \( \text{chr}_{\mathcal{P},\mathcal{Q}}(G)\) of $G$ is defined as the infimum of all ratios $a//b$ such that $G$ is \( (\mathcal{P},\mathcal{Q})\)-total $(a, b)$-list colorable. We give a direct proof of \( \chi_{ f,\mathcal{P},\mathcal{Q} }^{ \prime \prime } (G) = \text{chr}_{ \mathcal{P} ,\mathcal{Q} }(G)\) for all simple graphs $G$ and we present for some properties \( \mathcal{P} \) and \( \mathcal{Q} \) new bounds for the \( (\mathcal{P}, \mathcal{Q})\)-total chromatic number and for the \((\mathcal{P},\mathcal{Q})\)-choice ratio of a graph $G$.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2013, 33, 1; 167-179
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    3-Tuple Total Domination Number of Rook’s Graphs
    Autorzy:
    Pahlavsay, Behnaz
    Palezzato, Elisa
    Torielli, Michele
    Powiązania:
    https://bibliotekanauki.pl/articles/32361755.pdf
    Data publikacji:
    2022-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    k -tuple total domination
    Cartesian product of graphs
    rook’s graph
    Vizing’s conjecture
    Opis:
    A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S. The minimum size of a kTDS is called the k-tuple total dominating number and it is denoted by γ×k,t(G). We give a constructive proof of a general formula for γ×3,t(Kn□Km).
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 1; 15-37
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-35 z 35

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