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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:
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:
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:
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:
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:
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:
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:
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:
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:
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
Wyświetlanie 1-25 z 35

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