Vertex-critical [formula omitted]-free and vertex-critical (gem, co-gem)-free graphs

Abstract

A graph G is k-vertex-critical if χ(G)=k but χ(G−v)<k for all v∈V(G) where χ(G) denotes the chromatic number of G. We show that there are only finitely many k-vertex-critical (P3+ℓP1)-free graphs for all k≥1 and all ℓ≥0. Together with previous results, the only graphs H for which it is unknown if there are an infinite number of k-vertex-critical H-free graphs is H=(P4+ℓP1) for all ℓ≥1. We consider a restriction on the smallest open case, and show that there are only finitely many k-vertex-critical (gem, co-gem)-free graphs for all k, where gem=P4+P1¯. To do this, we show the stronger result that every vertex-critical (gem, co-gem)-free graph is either complete or a clique expansion of C5. This characterization allows us to give the complete list of all k-vertex-critical (gem, co-gem)-free graphs for all k≤16.