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arXiv · 2305.01923

The robust chromatic number of graphs

Gábor Bacsó·Balázs Patkós·Zsolt Tuza·Máté Vizer
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Abstract

A 1-removed subgraph $G_f$ of a graph $G=(V,E)$ is obtained by $(i)$ selecting at most one edge $f(v)$ for each vertex $v\in V$, such that $v\in f(v)\in E$ (the mapping $f:V\to E \cup \{\varnothing\}$ is allowed to be non-injective), and $(ii)$ deleting all the selected edges $f(v)$ from the edge set $E$ of $G$. Proper vertex colorings of 1-removed subgraphs proved to be a useful tool for earlier research on some Tur\'an-type problems. In this paper, we introduce a systematic investigation of the graph invariant 1-robust chromatic number, denoted as $\omega_1(G)$. This invariant is defined as the minimum chromatic number $\chi(G_f)$ among all 1-removed subgraphs $G_f$ of $G$. We also examine other standard graph invariants in a similar manner.

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Gábor Bacsó, Balázs Patkós, Zsolt Tuza, Máté Vizer. 2023-05-03. The robust chromatic number of graphs. https://arxiv.org/abs/2305.01923

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