Single conflict coloring, adaptable choosability and separation choosability

Abstract

We study relations between three interrelated notions of graph (list) coloring: single conflict coloring, adapted list coloring and choosability with separation (with $1$ overlapping color between lists of adjacent vertices), and their respective invariants single conflict chromatic number $\chi_{\nleftrightarrow}$, adaptable choosability $ch_{ad}$ and separation choosability $ch_{sep}$. We investigate graphs with small values of these invariants, and construct explicit families of graphs $G$ with $\chi_{\nleftrightarrow}(G) = ch_{ad}(G) > ch_{sep}(G)$, as well as where all three invariants are equal. Furthermore, we consider planar graphs and investigate for which triples $(a,b,c)$, there is a planar graph $G$ with $(ch_{sep}(G), ch_{ad}(G), \chi_{\nleftrightarrow}(G)) = (a,b,c)$. Throughout the paper we pose many questions on these graph coloring parameters, and discuss connections to related coloring invariants such as adapted coloring.