Acyclic colourings of graphs with obstructions
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2022
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| _version_ | 1866908826631929856 |
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| author | Chuet, Quentin Cohen, Johanne Pirot, François |
| author_facet | Chuet, Quentin Cohen, Johanne Pirot, François |
| contents | Given a graph $G$, a colouring of $G$ is \emph{acyclic} if it is a proper colouring of $G$ and every cycle contains at least three colours. Its acyclic chromatic number $χ_a(G)$ is the minimum~$k$ such that an acyclic $k$-colouring of $G$ exists. When $G$ has maximum degree $Δ$, it is known that $χ_a(G) = \mathcal {O}(Δ^{4/3})$ as $Δ\to \infty$, and that $χ_a(G) = \mathcal {O}(\sqrt{t} \cdot Δ)$ if in addition $G$ does not contain $K_{2,t}$ as a subgraph. We study the extremal value of the acyclic chromatic number in the class of graphs of maximum degree $Δ$ that do not contain some fixed subgraph $F$ on $t$ vertices. We establish that this extremal value is at most $\mathcal {O}(t^{8/3}Δ^{2/3})$ if $F$ is a tree, $\mathcal {O}(\sqrt{t} \cdot Δ)$ if $F$ is bipartite and can be made acyclic with the removal of one vertex, $2Δ+ \mathcal {O}(tΔ^{2/3})$ if $F$ is an even cycle of length at least $6$, and $\mathcal {O}(t^{1/4}Δ^{5/4})$ if $F=K_{3,t}$. Moreover, we exhibit an infinite family of obstructions $F$ that each induces a different asymptotic behaviour for this extremal value. This is obtained with the derivation of lower bounds that come from the analysis of the acyclic chromatic number of a random graph drawn from either $G(n,p)$ or $G(n,n,p)$, that we entirely determine up to a ${\rm polylog}(n)$ factor. As a byproduct, we can certify that most of our results are tight up to a $Δ^{\mathcal{O}(1/t)}$ factor. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_08417 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Acyclic colourings of graphs with obstructions Chuet, Quentin Cohen, Johanne Pirot, François Combinatorics Discrete Mathematics Given a graph $G$, a colouring of $G$ is \emph{acyclic} if it is a proper colouring of $G$ and every cycle contains at least three colours. Its acyclic chromatic number $χ_a(G)$ is the minimum~$k$ such that an acyclic $k$-colouring of $G$ exists. When $G$ has maximum degree $Δ$, it is known that $χ_a(G) = \mathcal {O}(Δ^{4/3})$ as $Δ\to \infty$, and that $χ_a(G) = \mathcal {O}(\sqrt{t} \cdot Δ)$ if in addition $G$ does not contain $K_{2,t}$ as a subgraph. We study the extremal value of the acyclic chromatic number in the class of graphs of maximum degree $Δ$ that do not contain some fixed subgraph $F$ on $t$ vertices. We establish that this extremal value is at most $\mathcal {O}(t^{8/3}Δ^{2/3})$ if $F$ is a tree, $\mathcal {O}(\sqrt{t} \cdot Δ)$ if $F$ is bipartite and can be made acyclic with the removal of one vertex, $2Δ+ \mathcal {O}(tΔ^{2/3})$ if $F$ is an even cycle of length at least $6$, and $\mathcal {O}(t^{1/4}Δ^{5/4})$ if $F=K_{3,t}$. Moreover, we exhibit an infinite family of obstructions $F$ that each induces a different asymptotic behaviour for this extremal value. This is obtained with the derivation of lower bounds that come from the analysis of the acyclic chromatic number of a random graph drawn from either $G(n,p)$ or $G(n,n,p)$, that we entirely determine up to a ${\rm polylog}(n)$ factor. As a byproduct, we can certify that most of our results are tight up to a $Δ^{\mathcal{O}(1/t)}$ factor. |
| title | Acyclic colourings of graphs with obstructions |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2211.08417 |