Infinite families of $k$-vertex-critical ($P_5$, $C_5$)-free graphs

TL;DR

This paper introduces new infinite families of $k$-vertex-critical graphs that are free of certain subgraphs, expanding the understanding of graph colorability and structure for all $k \, \geq \, 6$.

Contribution

It constructs the first known infinite families of $k$-vertex-critical $(P_5,C_5)$-free graphs for all $k \geq 6$, generalizing previous results.

Findings

Constructed infinite families for all $k \geq 6$

Generalizes known constructions for smaller $k$

Shows these graphs are $(2P_2,K_3+P_1,C_5)$-free

Abstract

A graph is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$. We construct a new infinite families of $k$-vertex-critical $(P_5,C_5)$-free graphs for all $k\ge 6$. Our construction generalizes 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 actually even more well-structured, being $(2P_2,K_3+P_1,C_5)$-free.