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    Wyświetlanie 1-14 z 14
    Tytuł:
    Equitable Total Coloring of Corona of Cubic Graphs
    Autorzy:
    Furmańczyk, Hanna
    Zuazua, Rita
    Powiązania:
    https://bibliotekanauki.pl/articles/32361758.pdf
    Data publikacji:
    2021-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    equitable coloring
    total coloring
    equitable total coloring
    cubic graphs
    Opis:
    The minimum number of total independent partition sets of V∪E of a graph G = (V, E) is called the total chromatic number of G, denoted by χ'′(G). If the difference between cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of V∪E is called the equitable total chromatic number, and is denoted by χ'′=(G). In this paper we consider equitable total coloring of coronas of cubic graphs, G◦H. It turns out that independently on the values of equitable total chromatic number of factors G and H, equitable total chromatic number of corona G◦H is equal to Δ(G◦H)+1. Thereby, we confirm Total Coloring Conjecture (TCC), posed by Behzad in 1964, and Equitable Total Coloring Conjecture (ETCC), posed by Wang in 2002, for coronas of cubic graphs. As a direct consequence we get that all coronas of cubic graphs are of Type 1.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 4; 1147-1163
    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:
    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:
    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:
    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:
    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:
    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:
    Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles
    Autorzy:
    Cranston, Daniel
    Powiązania:
    https://bibliotekanauki.pl/articles/743133.pdf
    Data publikacji:
    2009
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    list coloring
    edge coloring
    total coloring
    Vizing's Conjecture
    Opis:
    Let G be a planar graph with no two 3-cycles sharing an edge. We show that if Δ(G) ≥ 9, then χ'ₗ(G) = Δ(G) and χ''ₗ(G) = Δ(G)+1. We also show that if Δ(G) ≥ 6, then χ'ₗ(G) ≤ Δ(G)+1 and if Δ(G) ≥ 7, then χ''ₗ(G) ≤ Δ(G)+2. All of these results extend to graphs in the projective plane and when Δ(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with small degree-sum. For example, we prove that if G is a planar graph with no two 3-cycles sharing an edge and with Δ(G) ≥ 7, then G has an edge uv with d(u) ≤ 4 and d(u)+d(v) ≤ Δ(G)+2. All of our proofs yield linear-time algorithms that produce the desired colorings.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2009, 29, 1; 163-178
    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:
    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:
    An Improved Upper Bound on Neighbor Expanded Sum Distinguishing Index
    Autorzy:
    Vučković, Bojan
    Powiązania:
    https://bibliotekanauki.pl/articles/32083737.pdf
    Data publikacji:
    2020-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    general edge coloring
    total coloring
    neighbor sum distinguishing index
    Opis:
    A total k-weighting f of a graph G is an assignment of integers from the set {1, . . ., k} to the vertices and edges of G. We say that f is neighbor expanded sum distinguishing, or NESD for short, if Σw∈N(v) (f(vw) + f(w)) differs from Σw∈N(u)(f(uw) + f(w)) for every two adjacent vertices v and u of G. The neighbor expanded sum distinguishing index of G, denoted by egndiΣ(G), is the minimum positive integer k for which there exists an NESD weighting of G. An NESD weighting was introduced and investigated by Flandrin et al. (2017), where they conjectured that egndiΣ(G) ≤ 2 for any graph G. They examined some special classes of graphs, while proving that egndiΣ(G) ≤ χ(G) + 1. We improve this bound and show that egndiΣ(G) ≤ 3 for any graph G. We also show that the conjecture holds for all bipartite, 3-regular and 4-regular graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 1; 323-329
    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:
    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:
    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
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