Vertex-critical graphs in co-gem-free graphs

Abstract

A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$ and $(G,H)$-free if it contains no induced subgraph isomorphic to $G$ or $H$. We show that there are only finitely many $k$-vertex-critical (co-gem, $H$)-free graphs for all $k$ when $H$ is any graph of order $4$ by showing finiteness in the three remaining open cases, those are the cases when $H$ is $2P_2$, $K_3+P_1$, and $K_4$. For the first two cases we actually prove the stronger results: $\bullet$ There are only finitely many $k$-vertex-critical (co-gem, paw$+P_1$)-free graphs for all $k$ and that only finitely many $k$-vertex-critical (co-gem, paw$+P_1$)-free graphs for all $k\ge 1$. $\bullet$ There are only finitely many $k$-vertex-critical (co-gem, $P_5$, $P_3+cP_2$)-free graphs for all $k\ge 1$ and $c\ge 0$. To prove the latter result, we employ a novel application of Sperner's Theorem on the number of antichains in a partially ordered set. Our result for $K_4$ uses exhaustive computer search and is proved by showing the stronger result that every $(\text{co-gem, }K_4)$-free graph is $4$-colourable. Our results imply the existence of simple polynomial-time certifying algorithms to decide the $k$-colourability of (co-gem, $H$)-free graphs for all $k$ and all $H$ of order $4$ by searching the vertex-critical graphs as induced subgraphs.

Figures (2)

• Figure 1: All 11 nonisomorphic graphs of order four.
• Figure 2: All $4$-vertex-critical co-gem-free graphs.

Theorems & Definitions (33)

• Theorem 1.1: Chud4critical2020
• Theorem 1.2
• Lemma 2.1: Hoang2015
• Theorem 2.2: AbuadasCameronHoangSawada2022
• Theorem 2.3: Sperner's Theorem Sperner
• Theorem 2.4: AbuadasCameronHoangSawada2022
• Theorem 2.5: Chudnovsky4criticalconnected2020
• Corollary 2.6
• Theorem 3.1
• proof