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Tytuł:
Graph choosability and double list colorability
Autorzy:
Fanai, H. R.
Powiązania:
https://bibliotekanauki.pl/articles/255459.pdf
Data publikacji:
2010
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
list coloring
choosability
Opis:
In this paper, we give a sufficient condition for graph choosability, based on Combinatorial Nullstellensatz and a specific property, called "double list colorability", which means that there is a list assignment for which there are exactly two admissible colorings.
Źródło:
Opuscula Mathematica; 2010, 30, 3; 271-276
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
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:
Hajós theorem for list colorings of hypergraphs
Autorzy:
Benzaken, Claude
Gravier, Sylvain
Skrekovski, Riste
Powiązania:
https://bibliotekanauki.pl/articles/743401.pdf
Data publikacji:
2003
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
list-coloring
Hajós' construction
hypergraph
Opis:
A well-known theorem of Hajós claims that every graph with chromathic number greater than k can be constructed from disjoint copies of the complete graph $K_{k+1}$ by repeated application of three simple operations. This classical result has been extended in 1978 to colorings of hypergraphs by C. Benzaken and in 1996 to list-colorings of graphs by S. Gravier. In this note, we capture both variations to extend Hajós' theorem to list-colorings of hypergraphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2003, 23, 2; 207-213
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Generalized Sum List Colorings of Graphs
Autorzy:
Kemnitz, Arnfried
Marangio, Massimiliano
Voigt, Margit
Powiązania:
https://bibliotekanauki.pl/articles/31343297.pdf
Data publikacji:
2019-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
sum list coloring
sum choice number
generalized sum list coloring
additive hereditary graph property
Opis:
A (graph) property \( \mathcal{P} \) is a class of simple finite graphs closed under isomorphisms. In this paper we consider generalizations of sum list colorings of graphs with respect to properties \( \mathcal{P} \). If to each vertex $v$ of a graph $G$ a list $L(v)$ of colors is assigned, then in an \( (L, \mathcal{P} ) \)-coloring of $G$ every vertex obtains a color from its list and the subgraphs of $G$ induced by vertices of the same color are always in \( \mathcal{P} \). The \( \mathcal{P} \)-sum choice number \( X_{sc}^\mathcal{P} (G) \) of $G$ is the minimum of the sum of all list sizes such that, for any assignment $L$ of lists of colors with the given sizes, there is always an \( (L, \mathcal{P} ) \)-coloring of $G$. We state some basic results on monotonicity, give upper bounds on the \( \mathcal{P} \)-sum choice number of arbitrary graphs for several properties, and determine the \( \mathcal{P} \)-sum choice number of specific classes of graphs, namely, of all complete graphs, stars, paths, cycles, and all graphs of order at most 4.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 3; 689-703
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Defective choosability of graphs in surfaces
Autorzy:
Woodall, Douglas
Powiązania:
https://bibliotekanauki.pl/articles/743943.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
list coloring
defective coloring
minor-free graph
Opis:
It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ε, and k and d are positive integers such that k ≥ 3 and d is sufficiently large in terms of k and ε, then G is (k,d)*-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the known lower bound on d that suffices for this is reduced, and an analogous result is proved for list colorings (choosability). Also, the recent result of Cushing and Kierstead, that every planar graph is (4,1)*-choosable, is extended to $K_{3,3}$-minor-free and K₅-minor-free graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 3; 441-459
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:
Choice-Perfect Graphs
Autorzy:
Tuza, Zsolt
Powiązania:
https://bibliotekanauki.pl/articles/30146654.pdf
Data publikacji:
2013-03-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
graph coloring
list coloring
choice-perfect graph
Opis:
Given a graph $ G = (V,E) $ and a set $ L_v $ of admissible colors for each vertex $ v \in V $ (termed the list at $v$), a list coloring of $G$ is a (proper) vertex coloring $ \phi : V \rightarrow \bigcup \text{}_{v \in V} L_v $ such that $ \phi (v) \in L_v $ for all $ v \in V $ and $ \phi(u) \ne \phi(v) $ for all $ uv \in E $. If such a $ \phi $ exists, $G$ is said to be list colorable. The choice number of $G$ is the smallest natural number $k$ for which $G$ is list colorable whenever each list contains at least $k$ colors. In this note we initiate the study of graphs in which the choice number equals the clique number or the chromatic number in every induced subgraph. We call them choice-ω-perfect and choice-χ-perfect graphs, respectively. The main result of the paper states that the square of every cycle is choice-χ-perfect.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 1; 231-242
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:
A Note on the Equitable Choosability of Complete Bipartite Graphs
Autorzy:
Mudrock, Jeffrey A.
Chase, Madelynn
Thornburgh, Ezekiel
Kadera, Isaac
Wagstrom, Tim
Powiązania:
https://bibliotekanauki.pl/articles/32325306.pdf
Data publikacji:
2021-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
graph coloring
equitable coloring
list coloring
equitable choos-ability
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 said to be equitably k-choosable if an equitable L-coloring of G exists whenever L is a k-assignment for G. In this note we study the equitable choosability of complete bipartite graphs. A result of Kostochka, Pelsmajer, and West implies Kn,m is equitably k-choosable if k ≥ max{n, m} provided Kn,m ≠ K2l+1,2l+1. We prove Kn,m is equitably k-choosable if m ≤ ⌈ (m + n)/k⌉ (k − n) which gives Kn,m is equitably k-choosable for certain k satisfying k < max{n, m}. We also give a complete characterization of the equitable choosability of complete bipartite graphs that have a partite set of size at most 2.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 4; 1091-1101
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:
List coloring of complete multipartite graphs
Autorzy:
Vetrík, Tomáš
Powiązania:
https://bibliotekanauki.pl/articles/743641.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
list coloring
choice number
complete multipartite graph
Opis:
The choice number of a graph G is the smallest integer k such that for every assignment of a list L(v) of k colors to each vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from L(v). We present upper and lower bounds on the choice number of complete multipartite graphs with partite classes of equal sizes and complete r-partite graphs with r-1 partite classes of order two.
Źródło:
Discussiones Mathematicae Graph Theory; 2012, 32, 1; 31-37
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The list linear arboricity of planar graphs
Autorzy:
An, Xinhui
Wu, Baoyindureng
Powiązania:
https://bibliotekanauki.pl/articles/744447.pdf
Data publikacji:
2009
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
list coloring
linear arboricity
list linear arboricity
planar graph
Opis:
The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having Δ ≥ 13, or for any planar graph with Δ ≥ 7 and without i-cycles for some i ∈ {3,4,5}. We also prove that ⌈½Δ(G)⌉ ≤ lla(G) ≤ ⌈½(Δ(G)+1)⌉ for any planar graph having Δ ≥ 9.
Źródło:
Discussiones Mathematicae Graph Theory; 2009, 29, 3; 499-510
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On characterization of uniquely 3-list colorable complete multipartite graphs
Autorzy:
Zhao, Yancai
Shan, Erfang
Powiązania:
https://bibliotekanauki.pl/articles/744543.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
list coloring
complete multipartite graph
uniquely 3-list colorable graph
Opis:
For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_{2,2,r}$ r ∈ {4,5,6,7,8}, $K_{2,3,4}$, $K_{1*4,4}$, $K_{1*4,5}$, $K_{1*5,4}$. Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_{2,2,r}$ r ∈ {4,5,6,7,8}, the others have been proved not to be U3LC graphs. In this paper we first prove that $K_{2,2,8}$ is not U3LC graph, and thus as a direct corollary, $K_{2,2,r}$ (r = 4,5,6,7,8) are not U3LC graphs, and then the uniquely 3-list colorable complete multipartite graphs are characterized completely.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 1; 105-114
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Sum List Edge Colorings of Graphs
Autorzy:
Kemnitz, Arnfried
Marangio, Massimiliano
Voigt, Margit
Powiązania:
https://bibliotekanauki.pl/articles/31340809.pdf
Data publikacji:
2016-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
sum list edge coloring
sum choice index
sum list coloring
sum choice number
choice function
line graph
Opis:
Let $ G = (V,E) $ be a simple graph and for every edge $ \mathcal{e} \in E $ let $ L(e) $ be a set (list) of available colors. The graph $ G $ is called $L$-edge colorable if there is a proper edge coloring $ c $ of $ G $ with $ c(\mathcal{e} ) \in L( \mathcal{e} ) $ for all $ \mathcal{e} \in E $. A function $ f : E \rightarrow \mathbb{N} $ is called an edge choice function of $G$ and $G$ is said to be $f$-edge choosable if $G$ is $L$-edge colorable for every list assignment $L$ with $ |L( \mathcal{e} )| = f( \mathcal{e} ) $ for all $ \mathcal{e} \in E $. Set $ \text{size}(f) = \Sigma_{ \mathcal{e} \in E } f(e) $ and define the sum choice index $ \chi_{sc}^' (G) $ as the minimum of $ \text{size} (f) $ over all edge choice functions $f$ of $G$. There exists a greedy coloring of the edges of $G$ which leads to the upper bound $ \chi_{sc}^′ (G) \le 1/2 \Sigma_{ v \in V } d(v)^2 $. A graph is called sec-greedy if its sum choice index equals this upper bound. We present some general results on the sum choice index of graphs including a lower bound and we determine this index for several classes of graphs. Moreover, we present classes of sec-greedy graphs as well as all such graphs of order at most 5.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 3; 709-722
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Efficient list cost coloring of vertices and/or edges of bounded cyclicity graphs
Autorzy:
Giaro, Krzysztof
Kubale, Marek
Powiązania:
https://bibliotekanauki.pl/articles/744404.pdf
Data publikacji:
2009
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
cost coloring
dynamic programming
list coloring
NP-completeness
polynomial-time algorithm
Opis:
We consider a list cost coloring of vertices and edges in the model of vertex, edge, total and pseudototal coloring of graphs. We use a dynamic programming approach to derive polynomial-time algorithms for solving the above problems for trees. Then we generalize this approach to arbitrary graphs with bounded cyclomatic numbers and to their multicolorings.
Źródło:
Discussiones Mathematicae Graph Theory; 2009, 29, 2; 361-376
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
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