Colouring graphs from random lists
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866907860762361856 |
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| author | Hefetz, Dan Krivelevich, Michael |
| author_facet | Hefetz, Dan Krivelevich, Michael |
| contents | Given positive integers $k \leq m$ and a graph $G$, a family of lists $L = \{L(v) : v \in V(G)\}$ is said to be a random $(k,m)$-list-assignment if for every $v \in V(G)$ the list $L(v)$ is a subset of $\{1, \ldots, m\}$ of size $k$, chosen uniformly at random and independently of the choices of all other vertices. An $n$-vertex graph $G$ is said to be a.a.s. $(k,m)$-colourable if $\lim_{n \to \infty} \mathbb{P}(G \textrm{ is } L-colourable) = 1$, where $L$ is a random $(k,m)$-list-assignment. We prove that if $m \gg n^{1/k^2} Δ^{1/k}$ and $m \geq 3 k^2 Δ$, where $Δ$ is the maximum degree of $G$ and $k \geq 3$ is an integer, then $G$ is a.a.s. $(k,m)$-colourable. This is not far from being best possible, forms a continuation of the so-called palette sparsification results, and proves in a strong sense a conjecture of Casselgren. Additionally, we consider this problem under the additional assumption that $G$ is $H$-free for some graph $H$. For various graphs $H$, we estimate the smallest $m$ for which an $H$-free $n$-vertex graph $G$ is a.a.s. $(k,m)$-colourable. This extends and improves several results of Casselgren. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_09998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Colouring graphs from random lists Hefetz, Dan Krivelevich, Michael Combinatorics Given positive integers $k \leq m$ and a graph $G$, a family of lists $L = \{L(v) : v \in V(G)\}$ is said to be a random $(k,m)$-list-assignment if for every $v \in V(G)$ the list $L(v)$ is a subset of $\{1, \ldots, m\}$ of size $k$, chosen uniformly at random and independently of the choices of all other vertices. An $n$-vertex graph $G$ is said to be a.a.s. $(k,m)$-colourable if $\lim_{n \to \infty} \mathbb{P}(G \textrm{ is } L-colourable) = 1$, where $L$ is a random $(k,m)$-list-assignment. We prove that if $m \gg n^{1/k^2} Δ^{1/k}$ and $m \geq 3 k^2 Δ$, where $Δ$ is the maximum degree of $G$ and $k \geq 3$ is an integer, then $G$ is a.a.s. $(k,m)$-colourable. This is not far from being best possible, forms a continuation of the so-called palette sparsification results, and proves in a strong sense a conjecture of Casselgren. Additionally, we consider this problem under the additional assumption that $G$ is $H$-free for some graph $H$. For various graphs $H$, we estimate the smallest $m$ for which an $H$-free $n$-vertex graph $G$ is a.a.s. $(k,m)$-colourable. This extends and improves several results of Casselgren. |
| title | Colouring graphs from random lists |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.09998 |