Flexibility of triangle-free planar graphs

Dvořák, Zdeněk; Masařík, Tomáš; Musílek, Jan; Pangrác, Ondřej

doi:10.48550/arxiv.1902.02971

Cited by 3 publications

(7 citation statements)

References 12 publications

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“…The same is true for C 6 -free planar graphs [7]. For C 3 -free graphs, Dvořák, Masařík, Musílek, and Pangrác [4] showed that they are weighted ε-flexible for lists of size 4 and that this the result is tight. Surprisingly, the discharging proof in [4] is quite involved compared to the easy observation that C 3 -free planar graphs are 3-degenerate, which implies 4-choosability.…”

Section: Introductionmentioning

confidence: 76%

“…The concept of ε-flexibility was introduced by Dvořák, Norin, and Postle [6]. Subsequently, it was studied for various sub-classes of planar graphs, e.g., triangle-free [4], girth six [5], or C 4 -free [9]. Graphs of bounded maximum degree were subsequently characterized in terms of flexibility [1].…”

Section: Introductionmentioning

confidence: 99%

“…For C 3 -free graphs, Dvořák, Masařík, Musílek, and Pangrác [4] showed that they are weighted ε-flexible for lists of size 4 and that this the result is tight. Surprisingly, the discharging proof in [4] is quite involved compared to the easy observation that C 3 -free planar graphs are 3-degenerate, which implies 4-choosability. An analogous result holds for {C 3 , C 4 , C 5 }-free graphs, where list of size 3 are sufficient for weighted ε-flexibility and the result is tight [5].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Weak Flexibility in Planar Graphs

Lidický¹,

Masařík²,

Murphy³

et al. 2020

Preprint

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Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs [JGT 19']. In this new setting, each vertex v in some subset of V (G) has a request for a certain color r(v) in its list of colors L(v). The goal is to find an L coloring satisfying many, but not necessarily all, of the requests.The main studied question is whether there exists a universal constant ε > 0 such that any graph G in some graph class C satisfies at least ε proportion of the requests. More formally, for k > 0 the goal is to prove that for any graph G ∈ C on vertex set V , with any list assignment L of size k for each vertex, and for every R ⊆ V and a request vector (r(v) : v ∈ R, r(v) ∈ L(v)), there exists an L-coloring of G satisfying at least ε|R| requests. If this is true, then C is called ε-flexible for lists of size k.Choi et al. [arXiv 20'] introduced the notion of weak flexibility, where R = V . We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer b there exists ε(b) > 0 so that the class of planar graphs without K 4 , C 5 , C 6 , C 7 , B b is weakly ε(b)flexible for lists of size 4 (here K n , C n and B n are the complete graph, a cycle, and a book on n vertices, respectively). We also show that the class of planar graphs without K 4 , C 5 , C 6 , C 7 , B 5 is ε-flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable.

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Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Weak Flexibility in Planar Graphs

Lidický¹,

Masařík²,

Murphy³

et al. 2020

Preprint

Self Cite

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show abstract

“…Especially, in [2], Dvořák et al studied the 4-weighted flexibility of triangle-free planar graphs, and the result coincides with the choosability of triangle-free planar graphs. Later on, an interesting question is raised by Masařík [6] in the following.…”

Section: Introductionmentioning

confidence: 78%

Flexibility of planar graphs without $C_4$ and $C_5$

Yang,

Yang

2020

Preprint

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“…Apart from the original paper introducing flexibility [10], where some basic results in terms of maximum average degree were established, the main focus in flexibility research has been on planar graphs. In particular, for many subclasses G of planar graphs, there has been a vast effort to reduce the gap between the choosability of G and the list size needed for flexibility in G. As of now, some tight bounds on list sizes are known: namely, triangle-free planar graphs [8], {C 4 , C 5 }-free planar graphs [20], and {K 4 , C 5 , C 6 , C 7 , B 5 }-free 1 planar graphs [16] are flexibly 4-choosable, and planar graphs of girth 6 [9] are flexibly 3-choosable. For other subclasses G of planar graphs, only an upper-bound is known for the list size k required for G to be flexibly k-choosable [6,18]; see [6] for a comprehensive overview and a discussion of the related results.…”

Section: W(v C)mentioning

confidence: 99%

Flexible List Colorings in Graphs with Special Degeneracy Conditions

Bradshaw,

Masařík,

Stacho

2020

Preprint

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For a given ε > 0, we say that a graph G is ε-flexibly k-choosable if the following holds: for any assignment L of lists of size k on V (G), if a preferred color is requested at any set R of vertices, then at least ε|R| of these requests may be satisfied by some L-coloring. We consider flexible list colorings in several graph classes with certain special degeneracy conditions. We characterize the graphs of maximum degree ∆ that are ε-flexibly ∆-choosable for some ε = ε(∆) > 0, which answers a question of Dvořák, Norin, and Postle [List coloring with requests, JGT 2019]. We also show that graphs of treewidth 2 are 1 3 -flexibly 3-choosable, answering a question of I. Choi et al. [arXiv 2020], and we give conditions for list assignments by which graphs of treewidth k are 1 k+1 -flexibly (k + 1)-choosable. We show furthermore that graphs of treedepth k are 1 k -flexibly k-choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well-understood class of exceptions, three-connected non-regular graphs of maximum degree ∆ are flexibly (∆ − 1)-degenerate.

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Flexibility of triangle-free planar graphs

Cited by 3 publications

References 12 publications

On Weak Flexibility in Planar Graphs

On Weak Flexibility in Planar Graphs

Flexibility of planar graphs without $C_4$ and $C_5$

Flexible List Colorings in Graphs with Special Degeneracy Conditions

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