The Robust Chromatic Number of Graphs

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 \(f:V\rightarrow E \cup \{\varnothing \}\) 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 Turán-type problems. In this paper, we introduce a systematic investigation of the graph invariant 1-robust chromatic number, denoted as \(\chi _1(G)\). 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.