On choosability with separation of planar graphs with lists of different sizes

Kierstead, H. A.; Lidický, Bernard

doi:10.1016/j.disc.2015.01.008

Cited by 5 publications

(4 citation statements)

References 9 publications

(13 reference statements)

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“…In [10], Kratochvíl, Tuza, and Voigt showed that every planar graph is (4, 1)-choosable. This was later strengthened by Kierstead and Lidickỳ [9], who showed that the same result holds if we allow an independent set of vertices to have lists of size three. There exist planar graphs that are not (4, 4)-choosable: one such graph was given by Voigt in [14].…”

Section: Introductionmentioning

confidence: 82%

On the choosability with separation of planar graphs and its correspondence colouring analogue

Smith-Roberge¹

2022

Preprint

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We say G is (ℓ, k)-choosable if it admits an L-colouring for every (ℓ, k)-list assignment L. We prove that if G is a planar graph with (4, 2)-list assignment L and for every triangleAdditionally, we give a counterexample to the correspondence colouring analogue of (4, 2)-choosability for planar graphs. * We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [CGSD3 Grant No. 2020-547516].Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG) [CGSD3 Grant No. 2020-547516].

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Section: Introductionmentioning

confidence: 82%

On the choosability with separation of planar graphs and its correspondence colouring analogue

Smith-Roberge¹

2022

Preprint

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“…L that the lists of two adjacent vertices u and v satisfy |L(u) ∩ L(v)| ≤ a − c. Among the first results on the topic, a complexity dichotomy was presented [12] and general properties given [13]. Since then, a number of papers has considered choosability with separation of planar graphs, mainly for the case b = 1 [3,4,5,6,7,11,15]. A still open question for this class of graph is whether all planar graphs are (4, 1, 2)-choosable or not.…”

Section: Introductionmentioning

confidence: 99%

Choosability with Separation of Cycles and Outerplanar Graphs

Godin¹,

Togni²

2020

Preprint

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We consider the following list coloring with separation problem of graphs: Given a graph G and integers a, b, find the largest integer c such that for any list assignment L of G with |L(v)| ≤ a for any vertex v and |L(u) ∩ L(v)| ≤ c for any edge uv of G, there exists an assignment ϕ of sets of integers to the vertices of G such that ϕ(u) ⊂ L(u) and |ϕ(v)| = b for any vertex v and ϕ(u)∩ ϕ(v) = ∅ for any edge uv. Such a value of c is called the separation number of (G, a, b). We also study the variant called the free-separation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and free-separation number of the cycle and derive from them the free-separation number of a cactus. We also present a lower bound for the separation and free-separation numbers of outerplanar graphs of girth g ≥ 5.

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“…One refinement of choosability is called choosability with separation and has received recent attention [1,4,7,8,15] since it was defined by Kratochvíl, Tuza, and Voigt [10]. Let G be a graph and let s be a nonnegative integer called the separation parameter.…”

Section: Introductionmentioning

confidence: 99%

Choosability with union separation

Kumbhat

Moss

Stolee

2018

Discrete Mathematics

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Abstract. List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection of adjacent lists is sufficiently small. We introduce a new refinement, called choosability with union separation, where we require that the union of adjacent lists is sufficiently large. For t ≥ k, a (k, t)-list assignment is a list assignment L where |L(v)| ≥ k for all vertices v and |L(u) ∪ L(v)| ≥ t for all edges uv. A graph is (k, t)-choosable if there is a proper coloring for every (k, t)-list assignment. We explore this concept through examples of graphs that are not (k, t)-choosable, demonstrating sparsity conditions that imply a graph is (k, t)-choosable, and proving that all planar graphs are (3, 11)-choosable and (4, 9)-choosable.

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On choosability with separation of planar graphs with lists of different sizes

Cited by 5 publications

References 9 publications

On the choosability with separation of planar graphs and its correspondence colouring analogue

On the choosability with separation of planar graphs and its correspondence colouring analogue

Choosability with Separation of Cycles and Outerplanar Graphs

Choosability with union separation

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