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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:
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:
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:
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:
Generalized Fractional Total Colorings of Complete Graph
Autorzy:
Karafová, Gabriela
Powiązania:
https://bibliotekanauki.pl/articles/30145422.pdf
Data publikacji:
2013-09-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
fractional coloring
total coloring
complete graphs
Opis:
An additive and hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let $P$ and $Q$ be two additive and hereditary graph properties and let $r,s$ be integers such that $r\geq s$. Then an $\frac{r}{s}$-fractional $(P,Q)$-total coloring of a finite graph $G=(V,E)$ is a mapping $f$, which assigns an $s$-element subset of the set $\{1,2,...,r\}$ to each vertex and each edge, moreover, for any color $i$ all vertices of color $i$ induce a subgraph of property $P$, all edges of color $i$ induce a subgraph of property $Q$ and vertices and incident edges have assigned disjoint sets of colors. The minimum ratio $\frac{r}{s}$ of an $\frac{r}{s}$-fractional $(P,Q)$-total coloring of $G$ is called fractional $(P,Q)$-total chromatic number $\chi_{f,P,Q}^{''}(G)=\frac{r}{s}$. Let $k=$ sup$\{i:K_{i+1}\in P\}$ and $l=$ sup$\{i:K_{i+1}\in Q\}$. We show for a complete graph $K_{n}$ that if $l\geq k+2$ then $\chi_{f,P,Q}^{''}(K_{n})=\frac{n}{k+1}$ for a sufficiently large $n$.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 4; 665-676
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:
A Gallai-type equality for the total domination number of a graph
Autorzy:
Zhou, Sanming
Powiązania:
https://bibliotekanauki.pl/articles/744259.pdf
Data publikacji:
2004
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination number
total domination number
Gallai equality
Opis:
We prove the following Gallai-type equality
γₜ(G) + εₜ(G) = p
for any graph G with no isolated vertex, where p is the number of vertices of G, γₜ(G) is the total domination number of G, and εₜ(G) is the maximum integer s such that there exists a spanning forest F with s the number of pendant edges of F minus the number of star components of F.
Źródło:
Discussiones Mathematicae Graph Theory; 2004, 24, 3; 539-543
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On super edge-antimagic total labeling of subdivided stars
Autorzy:
Javaid, Muhammad
Powiązania:
https://bibliotekanauki.pl/articles/30148713.pdf
Data publikacji:
2014-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
subdivision of star
super ($a, d$)-EAT labeling
Opis:
In 1980, Enomoto et al. proposed the conjecture that every tree is a super ($a, 0$)-edge-antimagic total graph. In this paper, we give a partial support for the correctness of this conjecture by formulating some super ($a, d$)-edge-antimagic total labelings on a subclass of subdivided stars denoted by $T(n, n + 1, 2n + 1, 4n + 2, n_5, n_6, . . ., n_r)$ for different values of the edge-antimagic labeling parameter $d$, where $n ≥ 3$ is odd, $n_m = 2^{m−4}(4n+1)+1, r ≥ 5$ and $5 ≤ m ≤ r$.
Źródło:
Discussiones Mathematicae Graph Theory; 2014, 34, 4; 691-706
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:
Cubic Graphs with Total Domatic Number at Least Two
Autorzy:
Akbari, Saieed
Motiei, Mohammad
Mozaffari, Sahand
Yazdanbod, Sina
Powiązania:
https://bibliotekanauki.pl/articles/31342441.pdf
Data publikacji:
2018-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total domination
total domatic number
coupon coloring
Opis:
Let G be a graph with no isolated vertex. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we provide a criterion under which a cubic graph has total domatic number at least two.
Źródło:
Discussiones Mathematicae Graph Theory; 2018, 38, 1; 75-82
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 the Locating-Total Domination in Graphs
Autorzy:
Miller, Mirka
Rajan, R. Sundara
Jayagopal, R.
Rajasingh, Indra
Manuel, Paul
Powiązania:
https://bibliotekanauki.pl/articles/31341658.pdf
Data publikacji:
2017-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
dominating set
total dominating set
locating-dominating set
locating-total dominating set
regular graphs
Opis:
In this paper we obtain a sharp (improved) lower bound on the locating-total domination number of a graph, and show that the decision problem for the locating-total domination is NP-complete.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 3; 745-754
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:
Two Short Proofs on Total Domination
Autorzy:
Bickle, Allan
Powiązania:
https://bibliotekanauki.pl/articles/30146531.pdf
Data publikacji:
2013-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total domination
Opis:
A set of vertices of a graph G is a total dominating set if each vertex of G is adjacent to a vertex in the set. The total domination number of a graph γt (G) is the minimum size of a total dominating set. We provide a short proof of the result that γt (G) ≤ 2/3n for connected graphs with n ≥ 3 and a short characterization of the extremal graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 2; 457-459
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Note on Neighbor Expanded Sum Distinguishing Index
Autorzy:
Flandrin, Evelyne
Li, Hao
Marczyk, Antoni
Saclé, Jean-François
Woźniak, Mariusz
Powiązania:
https://bibliotekanauki.pl/articles/31342189.pdf
Data publikacji:
2017-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
general edge coloring
total coloring
neighbor-distinguishing index
neighbor sum distinguishing coloring
Opis:
A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set [k] = {1, . . ., k}. These colors can be used to distinguish the vertices of G. There are many possibilities of such a distinction. In this paper, we consider the sum of colors on incident edges and adjacent vertices.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 1; 29-37
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:
Hardness Results for Total Rainbow Connection of Graphs
Autorzy:
Chen, Lily
Huo, Bofeng
Ma, Yingbin
Powiązania:
https://bibliotekanauki.pl/articles/31340946.pdf
Data publikacji:
2016-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total rainbow connection
computational complexity
Opis:
A total-colored path is total rainbow if both its edges and internal vertices have distinct colors. The total rainbow connection number of a connected graph G, denoted by trc(G), is the smallest number of colors that are needed in a total-coloring of G in order to make G total rainbow connected, that is, any two vertices of G are connected by a total rainbow path. In this paper, we study the computational complexity of total rainbow connection of graphs. We show that deciding whether a given total-coloring of a graph G makes it total rainbow connected is NP-Complete. We also prove that given a graph G, deciding whether trc(G) = 3 is NP-Complete.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 2; 355-362
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On a Total Version of 1-2-3 Conjecture
Autorzy:
Baudon, Olivier
Hocquard, Hervé
Marczyk, Antoni
Pilśniak, Monika
Przybyło, Jakub
Woźniak, Mariusz
Powiązania:
https://bibliotekanauki.pl/articles/31348090.pdf
Data publikacji:
2020-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
neighbor sum distinguishing total coloring
general edge coloring
total coloring
neighbor-distinguishing index
neighbor full sum distinguishing total k -coloring
Opis:
A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set {1, . . ., k}. These colors can be used to distinguish adjacent vertices of G. There are many possibilities of such a distinction. In this paper, we focus on the one by the full sum of colors of a vertex, i.e., the sum of the color of the vertex, the colors on its incident edges and the colors on its adjacent vertices. This way of distinguishing vertices has similar properties to the method when we only use incident edge colors and to the corresponding 1-2-3 Conjecture.
Źródło:
Discussiones Mathematicae Graph Theory; 2020, 40, 4; 1175-1186
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
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:
Vertex-Distinguishing IE-Total Colorings of Complete Bipartite Graphs Km,N(m < n)
Autorzy:
Chen, Xiang’en
Gao, Yuping
Yao, Bing
Powiązania:
https://bibliotekanauki.pl/articles/30146641.pdf
Data publikacji:
2013-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
complete bipartite graphs
IE-total coloring
vertex-distinguishing IE-total coloring
vertex-distinguishing IE-total chromatic number
Opis:
Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C(u) be the set of colors of vertex u and edges incident to u under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χievt(G), and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. VDIET colorings of complete bipartite graphs Km,n(m < n) are discussed in this paper. Particularly, the VDIET chromatic numbers of Km,n(1 ≤ m ≤ 7, m < n) as well as complete graphs Kn are obtained.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 2; 289-306
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:
A characterization of locating-total domination edge critical graphs
Autorzy:
Blidia, Mostafa
Dali, Widad
Powiązania:
https://bibliotekanauki.pl/articles/743847.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
locating-domination
critical graph
Opis:
For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γₜ(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V(G)∖D, $N_G(u) ∩ D ≠ N_G(v) ∩ D$. The locating-total domination number $γ_L^t(G)$ is the minimum cardinality of a locating-total dominating set of G. A graph G is said to be a locating-total domination edge removal critical graph, or just a $γ_L^{t+}$-ER-critical graph, if $γ_L^t(G-e) > γ_L^t(G)$ for all e non-pendant edge of E. The purpose of this paper is to characterize the class of $γ_L^{t+}$-ER-critical graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 1; 197-202
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Generalized Fractional Total Colorings of Graphs
Autorzy:
Karafová, Gabriela
Soták, Roman
Powiązania:
https://bibliotekanauki.pl/articles/31339383.pdf
Data publikacji:
2015-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
fractional coloring
total coloring
automorphism group
Opis:
Let \( \mathcal{P} \) and \( \mathcal{Q} \) be additive and hereditary graph properties and let $r$, $s$ be integers such that $ r \ge s $. Then an $ r/s$-fractional (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of a finite graph $ G = (V, E) $ is a mapping $f$, which assigns an $s$-element subset of the set $ {1, 2, . . ., r}$ to each vertex and each edge, moreover, for any color $i$ all vertices of color $i$ induce a subgraph with property \( \mathcal{P} \), all edges of color $i$ induce a subgraph with property \( \mathcal{Q} \) and vertices and incident edges have been assigned disjoint sets of colors. The minimum ratio of an \( \frac{r}{s} \)-fractional (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of G is called fractional (\( \mathcal{P} \), \( \mathcal{Q} \))-total chromatic number \( \chi_{f, \mathcal{P} ,\mathcal{Q} }^{ \prime \prime } (G) = \frac{r}{s} \). We show in this paper that \( \chi_{f, \mathcal{P} ,\mathcal{Q} }^{ \prime \prime } \) of a graph \( G \) with \( o(V (G)) \) vertex orbits and \( o(E(G)) \) edge orbits can be found as a solution of a linear program with integer coefficients which consists only of \( o(V (G)) + o(E(G)) \) inequalities.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 3; 463-473
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total Protection of Lexicographic Product Graphs
Autorzy:
Martínez, Abel Cabrera
Rodríguez-Velázquez, Juan Alberto
Powiązania:
https://bibliotekanauki.pl/articles/32304140.pdf
Data publikacji:
2022-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total weak Roman domination
secure total domination
total domination
lexicographic product
Opis:
Given a graph G with vertex set V (G), a function f : V (G) → {0, 1, 2} is said to be a total dominating function if Σu∈N(v) f(u) > 0 for every v ∈ V (G), where N(v) denotes the open neighbourhood of v. Let Vi = {x ∈ V (G) : f(x) = i}. A total dominating function f is a total weak Roman dominating function if for every vertex v ∈ V0 there exists a vertex u ∈ N(v) ∩ (V1 ∪ V2) such that the function f′, defined by f′(v) = 1, f′(u) = f(u) − 1 and f′(x) = f(x) whenever x ∈ V (G) \ {u, v}, is a total dominating function as well. If f is a total weak Roman dominating function and V2 = ∅, then we say that f is a secure total dominating function. The weight of a function f is defined to be ω(f) = Σv∈V (G) f(v). The total weak Roman domination number (secure total domination number) of a graph G is the minimum weight among all total weak Roman dominating functions (secure total dominating functions) on G. In this article, we show that these two parameters coincide for lexicographic product graphs. Furthermore, we obtain closed formulae and tight bounds for these parameters in terms of invariants of the factor graphs involved in the product.
Źródło:
Discussiones Mathematicae Graph Theory; 2022, 42, 3; 967-984
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
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