Remarks on proper conflict-free colorings of graphs

Caro, Yair; Petruševski, Mirko; Škrekovski, Riste

doi:10.48550/arxiv.2203.01088

Cited by 3 publications

(7 citation statements)

References 30 publications

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“…In addition, every forest has treewidth at most 1, so Theorem 1.4 implies that every forest is conflict-free 3-choosable. It implies an earlier result in [7] stating that every forest is proper conflict-free 3-colorable. Note that this bound is tight since the 3-vertex path is not proper conflict-free 2-colorable.…”

Section: Our Resultssupporting

confidence: 53%

“…So Theorem 1.4 implies that every chordal graph with maximum degree at most ∆ is conflict-free (2∆ + 1)-choosable. It implies a result in [7] stating that every chordal graph with maximum degree at most ∆ is properly conflict-free (2∆ + 1)-colorable. In addition, every forest has treewidth at most 1, so Theorem 1.4 implies that every forest is conflict-free 3-choosable.…”

Section: Our Resultsmentioning

confidence: 95%

“…By combining the two notions of proper colorings and (not necessarily proper) colorings and the two notions of conflict-free colorings (with respect to open neighborhoods) and conflict-free colorings with respect to closed neighborhoods, there are four conflict-free-types of colorings studied in the literature (for example, [2,7,15,26]). For a graph G, we define χ pcf (G) and χ pcfc (G) to be the minimum k such that G admits a proper conflict-free k-coloring and a proper conflict-free k-coloring with respect to closed neighborhoods, respectively; we define χ icf (G) and χ icfc (G) to be the minimum k such that G admits a conflict-free k-coloring and a conflict-free k-coloring respect to closed neighborhoods, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

Liu¹

2022

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Proper conflict-free coloring is an intermediate notion between proper coloring of a graph and proper coloring of its square. Typical examples of graphs with large proper conflict-free chromatic number include (≤ 1)-subdivisions of graphs with large chromatic number. In this paper, we prove that a rough converse statement is true even for the list-coloring setting: for every graph H, there exists an integer c H such that every graph with no subdivision of H is (properly) conflict-free c H -choosable. This gives a (partial) answer of a question of Caro, Petruševski and Škrekovski. We also prove quantitatively better bounds for minor-closed families, implying some known results about (non-list) proper conflict-free coloring and odd coloring in the literature. Moreover, we prove that every graph with layered treewidth at most w is (properly) conflict-free (8w − 1)-choosable. This result applies to the class of (g, k)-planar graphs, which are graph classes whose coloring problems attract attentions recently.

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Section: Our Resultssupporting

confidence: 53%

Section: Our Resultsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

Liu¹

2022

Preprint

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“…Note that Theorem 1 is best possible as the conflict-free chromatic number is not bounded by the 1-strong colouring number [2]. Before proving Theorem 1, we highlight several noteworthy consequences.…”

mentioning

confidence: 95%

“…Dujmović, Morin, and Odak [6] proved a more general upper bound of O(k 5 ) for the odd chromatic number of k-planar graphs. See [1,2,4] for other results concerning these new graph parameters.…”

mentioning

confidence: 99%

Odd colourings, conflict-free colourings and strong colouring numbers

Hickingbotham¹

2022

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The odd chromatic number and the conflict-free chromatic number are new graph parameters introduced by Petruševski and Škrekovski Fabrici, Lužar, Rindošová andSoták [2022] respectively. In this note, we show that graphs with bounded 2-strong colouring number have bounded odd chromatic number and bounded conflict-free chromatic number. This implies that graph classes with bounded expansion have bounded odd chromatic number and bounded conflict-free chromatic number. Moreover, it follows by known results that the odd chromatic number and the conflict-free chromatic number of k-planar graphs is O(k) which improves a recent result of Dujmović, Morin and Odak [2022].

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Proper Conflict-free Coloring of Graphs with Large Maximum Degree

Cranston¹,

Liu²

2022

Preprint

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A proper coloring of a graph is conflict-free if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petruševski, and Škrekovski proved that every graph G has a proper conflict-free coloring with at most 5∆(G)/2 colors and conjectured that ∆(G) + 1 colors suffice for every connected graph G with ∆(G) 3. Our first main result is that even for list-coloring, 1.6550826∆(G) + ∆(G) colors suffice for every graph G with ∆(G) 10 9 ; we also prove slightly weaker bounds for all graphs with ∆(G) 750. These results follow from our more general framework on proper conflict-free list-coloring of a pair consisting of a graph G and a "conflict" hypergraph H. As another corollary of our results in this general framework, every graph has a proper ( √ 30 + o(1))∆(G) 1.5 -list-coloring such that every bi-chromatic component is a path on at most three vertices, where the number of colors is optimal up to a constant factor. Our proof uses a fairly new type of recursive counting argument called Rosenfeld counting, which is a variant of the Lovász Local Lemma or entropy compression.We also prove an asymptotically optimal result for a fractional analogue of our general framework for proper conflict-free coloring for pairs of a graph and a conflict hypergraph. A corollary states that every graph G has a fractional (1 + o(1))∆(G)-coloring such that every fractionally bi-chromatic component has at most two vertices. In particular, it implies that the fractional analogue of the conjecture of Caro et al. holds asymptotically in a strong sense.

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Remarks on proper conflict-free colorings of graphs

Cited by 3 publications

References 30 publications

Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

Odd colourings, conflict-free colourings and strong colouring numbers

Proper Conflict-free Coloring of Graphs with Large Maximum Degree

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