Sharp Dirac's theorem for DP‐critical graphs

Dvořák and Postle introduced DP‐coloring of simple graphs as a generalization of list‐coloring. They proved a Brooks' type theorem for DP‐coloring; and Bernshteyn, Kostochka, and Pron extended it to DP‐coloring of multigraphs. However, detailed structure, when a multigraph does not admit DP‐coloring, was not specified. In this note, we make this point clear and give the complete structure. This is also motivated by the relation to signed coloring of signed graphs.

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“…See [4] for more detailed results. Recently, there have been some works on DP-coloring; see [2,3,6,8].…”

Section: List-coloring and Dp-coloringmentioning

confidence: 99%

A note on a Brooks' type theorem for DP‐coloring

Kim

Ozeki

2018

Journal of Graph Theory

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“…See [4] for more detailed results. Recently, there are some works on DP-colorings; see [2,3,6,7,10].…”

Section: Dp-coloringmentioning

confidence: 99%

A sufficient condition for DP-4-colorability

Kim

Ozeki

2018

Discrete Mathematics

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DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each k ∈ {3, 4, 5, 6}, every planar graph without C k is 4-choosable. Furthermore, Jin, Kang, and Steffen [9] showed that for each k ∈ {3, 4, 5, 6}, every signed planar graph without C k is signed 4-choosable. In this paper, we show that for each k ∈ {3, 4, 5, 6}, every planar graph without C k is 4-DP-colorable, which is an extension of the above results.

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“…Much attention was drawn on this new coloring, see for example, [2,3,4,5,6,7,18,19,20,23,22]. We are interested in DP-coloring of planar graphs.…”

Section: Introductionmentioning

confidence: 99%

“…([21, 22]) A planar graph is DP-3-colorable if it has no cycles of length {4, 9, a, b}, where (a, b) ∈ {(5, 6),(5,7),(6,7),(6,8),(7,8)}.…”

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confidence: 99%

Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable

Yin

2019

Discrete Mathematics

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Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers d 0 and d 1 such that planar graphs without {4, 5}-cycles and d ∆ ≥ d 0 are 3-choosable and planar graphs without {4, 5, 6}-cycles and d ∆ ≥ d 1 are 3-choosable, where d ∆ is the smallest distance between triangles. They showed that 2 ≤ d 0 ≤ 4 and d 1 ≤ 3. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without {4, 5}-cycles and d ∆ ≥ 3 are DP-3-colorable, and (2) planar graphs without {4, 5, 6}-cycles and d ∆ ≥ 2 are DP-3-colorable. DP-coloring is a generalization of list-coloring, thus as a corollary, d 0 ≤ 3 and d 1 ≤ 2. We actually prove stronger statements that each pre-coloring on some cycles can be extended to the whole graph.

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Sharp Dirac's theorem for DP‐critical graphs

Cited by 34 publications

References 15 publications

A note on a Brooks' type theorem for DP‐coloring

A note on a Brooks' type theorem for DP‐coloring

A sufficient condition for DP-4-colorability

Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable

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