Critical ($P_5$,bull)-free graphs

TL;DR

This paper proves that there are finitely many 5-vertex-critical graphs that do not contain an induced path of length five or a bull graph, advancing understanding of graph structure constraints.

Contribution

It establishes the finiteness of 5-vertex-critical ($P_5$,bull)-free graphs, a new result in the study of graph classes defined by forbidden induced subgraphs.

Findings

Finiteness of 5-vertex-critical ($P_5$,bull)-free graphs.

Characterization of forbidden induced subgraphs.

Implications for graph coloring and structure theory.

Abstract

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_t$ be the path and the cycle on $t$ vertices, respectively. A bull is the graph obtained from a triangle with two disjoint pendant edges. In this paper, we show that there are finitely many 5-vertex-critical ($P_5$,bull)-free graphs.