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

Abstract

Proper conflict-free coloring is an intermediate notion between proper coloring of a graph and proper coloring of its square. It is a proper coloring such that for every non-isolated vertex, there exists a color appearing exactly once in its (open) neighborhood. Typical examples of graphs with large proper conflict-free chromatic number include graphs with large chromatic number and bipartite graphs isomorphic to the $1$-subdivision of graphs with large chromatic number. In this paper, we prove two rough converse statements that hold even in the list-coloring setting. The first is for sparse graphs: 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. The second applies to dense graphs: every graph with large conflict-free choice number either contains a large complete graph as an odd minor or contains a bipartite induced subgraph that has large conflict-free choice number. These give two incomparable (partial) answers of a question of Caro, Petruševski and Škrekovski. We also prove quantitatively better bounds for minor-closed families, implying some known results about 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 $(g,k)$-planar graphs, which are graphs whose coloring problems have attracted attention recently.

Hickingbotham recently independently announced a paper (arXiv:2203.10402) proving a result similar to the ones in this paper. Please see the notes at the end of this paper for details. v2: add results for odd minors, which applies to graphs with unbounded degeneracy, and change the title of the paper

References in corpus (4)

• Proper conflict-free and unique-maximum colorings of planar graphs with respect to neighborhoods
• A Note on Odd Colorings of 1-Planar Graphs
• Separating layered treewidth and row treewidth
• Extremal functions for sparse minors

Cited by in corpus (6)

• Proper Conflict-free Coloring of Graphs with Large Maximum Degree
• Asymptotically Optimal Proper Conflict-Free Colouring
• The proper conflict-free $k$-coloring problem and the odd $k$-coloring problem are NP-complete on bipartite graphs
• Boundedness for proper conflict-free and odd colorings
• Weak diameter choosability of graphs with an excluded minor
• Relaxation of Wegner's Planar Graph Conjecture for maximum degree 4