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1
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Choice-Perfect Graphs

100%
Tuza Z.
Discussiones Mathematicae Graph Theory
|
2013
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tom 33
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nr 1
231-242
EN
Given a graph G = (V,E) and a set Lv of admissible colors for each vertex v ∈ V (termed the list at v), a list coloring of G is a (proper) vertex coloring ϕ : V → S v2V Lv such that ϕ(v) ∈ Lv for all v ∈ V and ϕ(u) 6= ϕ(v) for all uv ∈ E. If such a ϕ 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.
2
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Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles

100%
Cranston D.
Discussiones Mathematicae Graph Theory
|
2009
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tom 29
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nr 1
163-178
EN
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.
3
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The list linear arboricity of planar graphs

100%
An X., Wu B.
Discussiones Mathematicae Graph Theory
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2009
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tom 29
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nr 3
499-510
EN
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.
4
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On characterization of uniquely 3-list colorable complete multipartite graphs

100%
Zhao Y., Shan E.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
|
nr 1
105-114
EN
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.
5
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List coloring of complete multipartite graphs

100%
Vetrík T.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
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nr 1
31-37
EN
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.
6
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On choosability of complete multipartite graphs $K_{4,3*t,2*(k-2t-2),1*(t+1)}$

100%
Zheng G.-P., Shen Y.-F., Chen Z.-L., Lv J.-F.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
|
nr 2
237-244
EN
A graph G is said to be chromatic-choosable if ch(G) = χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba's conjecture is true if and only if it is true for complete multipartite graphs. In this paper we show that Ohba's conjecture is true for complete multipartite graphs $K_{4,3*t,2*(k-2t-2),1*(t+1)}$ for all integers t ≥ 1 and k ≥ 2t+2, that is, $ch(K_{4,3*t,2*(k-2t-2),1*(t+1)}) = k$, which extends the results $ch(K_{4,3,2*(k-4),1*2}) = k$ given by Shen et al. (Discrete Math. 308 (2008) 136-143), and $ch(K_{4,3*2,2*(k-6),1*3}) = k$ given by He et al. (Discrete Math. 308 (2008) 5871-5877).
7
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Defective choosability of graphs in surfaces

100%
Woodall D.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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nr 3
441-459
EN
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.
8
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A note on total colorings of planar graphs without 4-cycles

100%
Wang P., Wu J.-L.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
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nr 1
125-135
EN
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)}.
9
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Efficient list cost coloring of vertices and/or edges of bounded cyclicity graphs

88%
Giaro K., Kubale M.
Discussiones Mathematicae Graph Theory
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2009
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tom 29
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nr 2
361-376
EN
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.
10
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Graph colorings with local constraints - a survey

75%
Tuza Z.
Discussiones Mathematicae Graph Theory
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1997
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tom 17
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nr 2
161-228
EN
We survey the literature on those variants of the chromatic number problem where not only a proper coloring has to be found (i.e., adjacent vertices must not receive the same color) but some further local restrictions are imposed on the color assignment. Mostly, the list colorings and the precoloring extensions are considered. In one of the most general formulations, a graph G = (V,E), sets L(v) of admissible colors, and natural numbers $c_v$ for the vertices v ∈ V are given, and the question is whether there can be chosen a subset C(v) ⊆ L(v) of cardinality $c_v$ for each vertex in such a way that the sets C(v),C(v') are disjoint for each pair v,v' of adjacent vertices. The particular case of constant |L(v)| with $c_v$ = 1 for all v ∈ V leads to the concept of choice number, a graph parameter showing unexpectedly different behavior compared to the chromatic number, despite these two invariants have nearly the same value for almost all graphs. To illustrate typical techniques, some of the proofs are sketched.
11
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Fractional (P,Q)-Total List Colorings of Graphs

63%
Kemnitz A., Mihók P., Voigt M.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 1
167-179
EN
Let r, s ∈ N, r ≥ s, and P and 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 Zr such that for each color i, 0 ≤ i ≤ r − 1, the vertices colored by subsets containing i induce a subgraph of G with property P, the edges colored by subsets containing i induce a subgraph of G with property Q, and color sets of incident vertices and edges are disjoint. The fractional (P,Q)-total chromatic number χ′′ f,P,Q(G) of G is defined as the infimum of all ratios r/s such that G has a (P,Q)-total (r, s)-coloring. A (P,Q)-total independent set T = VT ∪ET ⊆ V ∪E is the union of a set VT of vertices and a set ET of edges of G such that for the graphs induced by the sets VT and ET it holds that G[VT ] ∈ P, G[ET ] ∈ Q, and G[VT ] and G[ET ] are disjoint. Let TP,Q be the set of all (P,Q)-total independent sets of G. Let L(x) be a set of admissible colors for every element x ∈ V ∪ E. The graph G is called (P,Q)-total (a, b)-list colorable if for each list assignment L with |L(x)| = a for all x ∈ V ∪E it is possible to choose a subset C(x) ⊆ L(x) with |C(x)| = b for all x ∈ V ∪ E such that the set Ti which is defined by Ti = {x ∈ V ∪ E : i ∈ C(x)} belongs to TP,Q for every color i. The (P,Q)- choice ratio chrP,Q(G) of G is defined as the infimum of all ratios a/b such that G is (P,Q)-total (a, b)-list colorable. We give a direct proof of χ′′ f,P,Q(G) = chrP,Q(G) for all simple graphs G and we present for some properties P and Q new bounds for the (P,Q)-total chromatic number and for the (P,Q)-choice ratio of a graph G.
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