Infinite Families of k-Vertex-Critical ( $$P_5$$ , $$C_5$$ )-Free Graphs

Abstract

A graph is k-vertex-critical if \(\chi (G)=k\) but \(\chi (G-v)<k\) for all \(v\in V(G)\). We construct new infinite families of k-vertex-critical \((P_5,C_5)\)-free graphs for all \(k\ge 6\). Our construction generalises known constructions for 4-vertex-critical \(P_7\)-free graphs and 5-vertex-critical \(P_5\)-free graphs and is in contrast to the fact that there are only finitely many 5-vertex-critical \((P_5,C_5)\)-free graphs. In fact, our construction is even more well-structured, being \((2P_2,K_3+P_1,C_5)\)-free.