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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.08393 |
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| _version_ | 1866916359533756416 |
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| author | Bi, Richard Bradshaw, Peter |
| author_facet | Bi, Richard Bradshaw, Peter |
| contents | We consider the flexible list coloring problem, in which we have a graph $G$, a color list assignment $L:V(G) \rightarrow 2^{\mathbb N}$, and a set $U \subseteq V(G)$ of vertices such that each $u \in U$ has a preferred color $p(u) \in L(u)$. Given a constant $\varepsilon > 0$, the problem asks for an $L$-coloring of $G$ in which at least $\varepsilon |U|$ vertices in $U$ receive their preferred color. We use a method of reducible subgraphs to approach this problem. We develop a vertex-partitioning tool that, when used with a new reducible subgraph framework, allows us to define large reducible subgraphs. Using this new tool, we show that if $G$ has maximum average degree less than $\frac{11}{3}$, a list $L(v)$ of size $4$ at each $v \in V(G)$, and a set $U \subseteq V(G)$ of vertices with preferred colors, then there exists an $L$-coloring of $G$ for which at least $2^{-145} |U|$ vertices of $U$ receive their preferred color. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08393 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graphs of maximum average degree less than $\frac {11}{3}$ are flexibly $4$-choosable Bi, Richard Bradshaw, Peter Combinatorics 05C15 We consider the flexible list coloring problem, in which we have a graph $G$, a color list assignment $L:V(G) \rightarrow 2^{\mathbb N}$, and a set $U \subseteq V(G)$ of vertices such that each $u \in U$ has a preferred color $p(u) \in L(u)$. Given a constant $\varepsilon > 0$, the problem asks for an $L$-coloring of $G$ in which at least $\varepsilon |U|$ vertices in $U$ receive their preferred color. We use a method of reducible subgraphs to approach this problem. We develop a vertex-partitioning tool that, when used with a new reducible subgraph framework, allows us to define large reducible subgraphs. Using this new tool, we show that if $G$ has maximum average degree less than $\frac{11}{3}$, a list $L(v)$ of size $4$ at each $v \in V(G)$, and a set $U \subseteq V(G)$ of vertices with preferred colors, then there exists an $L$-coloring of $G$ for which at least $2^{-145} |U|$ vertices of $U$ receive their preferred color. |
| title | Graphs of maximum average degree less than $\frac {11}{3}$ are flexibly $4$-choosable |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2408.08393 |