On Gy\'arf\'as' Path-Colour Problem

TL;DR

This paper investigates Gyárfás' path-coloring conjecture, constructs graphs with specific properties, and explores bounds on chromatic number in graphs with forbidden subgraphs, extending the understanding of graph coloring related to path-induced subgraphs.

Contribution

The paper provides a constructive proof that graphs with bounded path chromatic number can have arbitrarily large chromatic number, and establishes new bounds for graphs with forbidden induced subgraphs.

Findings

Existence of graphs with bounded r(G) and arbitrarily large chromatic number.

Bound on chromatic number for K_{1,t}-free graphs in terms of r(G).

Path-perfect graphs generalize perfect graphs with specific forbidden subgraph conditions.

Abstract

In their 1997 paper titled ``Fruit Salad", Gy\'{a}rf\'{a}s posed the following conjecture: there exists a constant $k$ such that if each path of a graph spans a $3$-colourable subgraph, then the graph is $k$-colourable. It is noted that $k=4$ might suffice. Let $r(G)$ be the maximum chromatic number of any subgraph $H$ of $G$ where $H$ is spanned by a path. The only progress on this conjecture comes from Randerath and Schiermeyer in 2002, who proved that if $G$ is an $n$ vertex graph, then $\chi(G) \leq r(G)\log_{\frac{8}{7}}(n)$. We prove that for all natural numbers $r$, there exists a graph $G$ with $r(G)\leq r$ and $\chi(G)\geq \lfloor\frac{3r}{2}\rfloor -1$. Hence, for all constants $k$ there exists a graph with $\chi - r > k$. Our proof is constructive. We also study this problem in graphs with a forbidden induced subgraph. We show that if $G$ is $K_{1,t}$-free, for $t\geq 4$, then $\chi(G) \leq (t-1)(r(G)+\binom{t-1}{2}-3)$. If $G$ is claw-free, then we prove $\chi(G) \leq 2r(G)$. Additionally, the graphs $G$ where every induced subgraph $G'$ of $G$ satisfy $\chi(G') = r(G')$ are considered. We call such graphs path-perfect, as this class generalizes perfect graphs. We prove that if $H$ is a forest with at most $4$ vertices other than the claw, then every $H$-free graph $G$ has $\chi(G) \leq r(G)+1$. We also prove that if $H$ is additionally not isomorphic to $2K_2$ or $K_2+2K_1$, then all $H$-free graphs are path-perfect.