Critical $(P_5,dart)$-Free Graphs

TL;DR

This paper proves that there are finitely many $k$-vertex-critical $(P_5,dart)$-free graphs for all $k$, characterizes them for small $k$, and provides a polynomial-time algorithm for $k$-colorability in this class.

Contribution

It establishes finiteness, characterizes small critical graphs using computational methods, and develops a certifying algorithm for $k$-colorability of $(P_5,dart)$-free graphs.

Findings

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

Complete characterization of such graphs for $k=5,6,7$.

Existence of a polynomial-time certifying $k$-colorability algorithm.

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 $H_1$ nor $H_2$. Let $P_t$ be the path on $t$ vertices. A dart is the graph obtained from a diamond by adding a new vertex and making it adjacent to exactly one vertex with degree 3 in the diamond. In this paper, we show that there are finitely many $k$-vertex-critical $(P_5,dart)$-free graphs for $k \ge 1$ To prove these results, we use induction on $k$ and perform a careful structural analysis via Strong Perfect Graph Theorem combined with the pigeonhole principle based on the properties of vertex-critical graphs. Moreover, for $k \in \{5, 6, 7\}$ we characterize all $k$-vertex-critical $(P_5,dart)$-free graphs using a computer generation algorithm. Our results imply the existence of a polynomial-time certifying algorithm to decide the $k$-colorability of $(P_5,dart)$-free graphs for $k \ge 1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph.