A refinement on the structure of vertex-critical ($P_5$, gem)-free graphs

TL;DR

This paper refines the structural understanding of vertex-critical ($P_5$, gem)-free graphs, proving finiteness for all $k$ and enabling polynomial-time certifying algorithms for their $k$-colorability.

Contribution

It provides a new, stronger proof of finiteness, refines graph structure, and lists all 6- and 7-vertex-critical graphs, facilitating efficient coloring algorithms.

Findings

Finiteness of $k$-vertex-critical ($P_5$, gem)-free graphs for all $k$

Complete enumeration of 6- and 7-vertex-critical graphs

Polynomial-time certifying algorithms for $k$-colorability

Abstract

We give a new, stronger proof that there are only finitely many $k$-vertex-critical ($P_5$,~gem)-free graphs for all $k$. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all $6$- and $7$-vertex-critical $(P_5$, gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the $k$-colourability of $(P_5$, gem)-free graphs for all $k$ where the certificate is either a $k$-colouring or a $(k+1)$-vertex-critical induced subgraph. Our complete lists for $k\le 7$ allow for the implementation of these algorithms for all $k\le 6$.