A counterexample to a conjecture on facial unique-maximal colorings

Lidický, Bernard; Messerschmidt, Kacy; Škrekovski, Riste

doi:10.1016/j.dam.2017.11.037

Cited by 5 publications

(3 citation statements)

References 6 publications

(12 reference statements)

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“…If this is true, maybe it could be extended to maximum degree 3 in G[X]. The connected plane graph H with χ fum (H) > 4 found in [5] has minimum degree 4 and two vertices of degree five. The construction could be disconnected, which gives a 4-regular graph.…”

Section: Discussionmentioning

confidence: 99%

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Facial unique-maximum colorings of plane graphs with restriction on big vertices

Lidický

Messerschmidt

Škrekovski

2019

Discrete Mathematics

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A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and Göring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial uniquemaximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially uniquemaximum 4-colorable. This improves a previous result for subcubic plane graphs by Andova, Lidický, Lužar, and Škrekovski (2018). We conclude the paper by proposing some problems.

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Section: Discussionmentioning

confidence: 99%

“…This conjecture was disproven in the general case by the authors [5]. Fabrici and Göring [4] proved that for any plane graph G, χ fum (G) ≤ 6, while Wendland [8] improved the upper bound to 5.…”

Section: Introductionmentioning

confidence: 99%

Facial unique-maximum colorings of plane graphs with restriction on big vertices

Lidický

Messerschmidt

Škrekovski

2019

Discrete Mathematics

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“…Wendland [26] improved the bound of 6 colors from [12] to 5. Later, it turned out that there is an infinite family of examples attaining it [16], and so the upper bound is tight.…”

Section: Preliminariesmentioning

confidence: 99%

Proper conflict-free and unique-maximum colorings of planar graphs with respect to neighborhoods

Fabrici¹,

Lužar²,

Rindošová³

et al. 2022

Preprint

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A conflict-free coloring of a graph with respect to open (resp., closed) neighborhood is a coloring of vertices such that for every vertex there is a color appearing exactly once in its open (resp., closed) neighborhood. Similarly, a unique-maximum coloring of a graph with respect to open (resp., closed) neighborhood is a coloring of vertices such that for every vertex the maximum color appearing in its open (resp., closed) neighborhood appears exactly once. There is a vast amount of literature on both notions where the colorings need not be proper, i.e., adjacent vertices are allowed to have the same color.In this paper, we initiate a study of both colorings in the proper settings with the focus given mainly to planar graphs. We establish upper bounds for the number of colors in the class of planar graphs for all considered colorings and provide constructions of planar graphs attaining relatively high values of the corresponding chromatic numbers. As a main result, we prove that every planar graph admits a proper unique-maximum coloring with respect to open neighborhood with at most 10 colors, and give examples of planar graphs needing at least 6 colors for such a coloring. We also establish tight upper bounds for outerplanar graphs. Finally, we provide several new bounds also for the improper setting of considered colorings.

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