The strong fractional choice number of triangle-free planar graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910023036174336 |
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| author | Hu, Xiaolan Xu, Rongxing |
| author_facet | Hu, Xiaolan Xu, Rongxing |
| contents | Let $a,b$ be positive integers with $a\ge b$. A graph $G$ is $(a,b)$-choosable if, for every assignment of lists $L(v)$ of size $a$ to the vertices of $G$, there exists a choice of subsets $C(v)\subseteq L(v)$ with $|C(v)|=b$ for each $v$ such that $C(u)\cap C(v)=\emptyset$ whenever $uv\in E(G)$. We show that every triangle-free planar graph is $(15m,4m)$-choosable for any positive integer $m$. As an immediate consequence, the strong fractional choice number of triangle-free planar graphs is at most $15/4$. This appears to be the first non-trivial upper bound on this parameter for this class of graphs. In particular, the case $m=1$ answers affirmatively a question posed by Jiang and Zhu in [J.~Combin.\ Theory Ser.~B, 2019]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The strong fractional choice number of triangle-free planar graphs Hu, Xiaolan Xu, Rongxing Combinatorics Let $a,b$ be positive integers with $a\ge b$. A graph $G$ is $(a,b)$-choosable if, for every assignment of lists $L(v)$ of size $a$ to the vertices of $G$, there exists a choice of subsets $C(v)\subseteq L(v)$ with $|C(v)|=b$ for each $v$ such that $C(u)\cap C(v)=\emptyset$ whenever $uv\in E(G)$. We show that every triangle-free planar graph is $(15m,4m)$-choosable for any positive integer $m$. As an immediate consequence, the strong fractional choice number of triangle-free planar graphs is at most $15/4$. This appears to be the first non-trivial upper bound on this parameter for this class of graphs. In particular, the case $m=1$ answers affirmatively a question posed by Jiang and Zhu in [J.~Combin.\ Theory Ser.~B, 2019]. |
| title | The strong fractional choice number of triangle-free planar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.13970 |