Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth
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| Format: | Preprint |
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2022
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| _version_ | 1866909425796644864 |
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| author | Liu, Chun-Hung |
| author_facet | Liu, Chun-Hung |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_12248 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth Liu, Chun-Hung Combinatorics 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. |
| title | Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2203.12248 |