On Gyárfás' Path-Colour Problem
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866909658642382848 |
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| author | Cameron, Ben Clow, Alexander |
| author_facet | Cameron, Ben Clow, Alexander |
| contents | In their 1997 paper titled ``Fruit Salad", Gyárfás 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 $χ(G) \leq r(G)\log_{\frac{8}{7}}(n)$.
We prove that for all natural numbers $r$, there exists a graph $G$ with $r(G)\leq r$ and $χ(G)\geq \lfloor\frac{3r}{2}\rfloor -1$. Hence, for all constants $k$ there exists a graph with $χ- r > k$. 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 $χ(G) \leq (t-1)(r(G)+\binom{t-1}{2}-3)$. If $G$ is claw-free, then we prove $χ(G) \leq 2r(G)$. Additionally, the graphs $G$ where every induced subgraph $G'$ of $G$ satisfy $χ(G') = r(G')$ 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 $χ(G) \leq r(G)+1$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Gyárfás' Path-Colour Problem Cameron, Ben Clow, Alexander Combinatorics 05C15, 05C38 In their 1997 paper titled ``Fruit Salad", Gyárfás 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 $χ(G) \leq r(G)\log_{\frac{8}{7}}(n)$. We prove that for all natural numbers $r$, there exists a graph $G$ with $r(G)\leq r$ and $χ(G)\geq \lfloor\frac{3r}{2}\rfloor -1$. Hence, for all constants $k$ there exists a graph with $χ- r > k$. 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 $χ(G) \leq (t-1)(r(G)+\binom{t-1}{2}-3)$. If $G$ is claw-free, then we prove $χ(G) \leq 2r(G)$. Additionally, the graphs $G$ where every induced subgraph $G'$ of $G$ satisfy $χ(G') = r(G')$ 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 $χ(G) \leq r(G)+1$. 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. |
| title | On Gyárfás' Path-Colour Problem |
| topic | Combinatorics 05C15, 05C38 |
| url | https://arxiv.org/abs/2506.19100 |