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

Abstract

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.

Remarks on proper conflict-free colorings of graphs

“…The recent work of Fabrici et al [18], where the notion of 'proper conflict-free coloring' was introduced, focuses on planar and outerplanar graphs. Among other things, [18] contains a 'proof from the Book' of the fact that every planar graph is PCF 8-colorable (cf. Theorem 5.3).…”

“…For graphs, Cheilaris [7] studied the CF coloring with respect to neighborhoods, that is, the coloring in which for every non-isolated vertex x there is a color that occurs exactly once in the (open) neighborhood N(x), and proved the upper bound 2 √ n for the CF chromatic number of a graph of order n. For more on not necessarily proper CF colorings see, e.g., [9,19,23,28,33]. Quite recently, Fabrici et al [18] initiated a study of proper conflict-free colorings with respect to neighborhoods while focusing mainly on planar and outerplanar graphs. In fact, combining the coloring notions of 'conflict-free' and 'proper' is only natural as the former was initially introduced (in the hypergraph setting) in order to generalize the latter.…”

Odd colourings, conflict-free colourings and strong colouring numbers

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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.

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“…Motivated by connections to hypergraph colouring, the odd chromatic number and the conflict-free chromatic number were recently introduced by Petruševski and Škrekovski [19] and Fabrici, Lužar, Rindošová, and Soták [10] respectively. These parameters have gained significant traction with a particular focus on determining a tight upper bound for planar graphs.…”

“…Petr and Portier [18] improved this upper bound to 8. For conflict-free colourings, Fabrici et al [10] proved a matching upper bound of 8 for planar graphs. For proper minor-closed classes, a result of Cranston, Lafferty, and Song [5] implies that the odd chromatic number of K t -minor free graphs is O(t √ log t).…”