Equitably Coloring Planar and Outerplanar Graphs
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| Format: | Preprint |
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2025
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| _version_ | 1866912598441590784 |
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| author | Cranston, Daniel W. Mahmoud, Reem |
| author_facet | Cranston, Daniel W. Mahmoud, Reem |
| contents | A proper $s$-coloring of an $n$-vertex graph is \emph{equitable} if every color class has size $\lfloor{n/s}\rfloor$ or $\lceil{n/s}\rceil$. A necessary condition to have an equitable $s$-coloring is that every vertex $v$ appears in an independent set of size at least $\lfloor{n/s}\rfloor$. That is $\min_{v\in V(G)}α_v\ge \lfloor{n/s}\rfloor$. Various authors showed that when $G$ is a tree and $s\ge 3$ this obvious necessary condition is also sufficient. Kierstead, Kostochka, and Xiang asked whether this result holds more generally for all outerplanar graphs. We show that the answer is No when $s=3$, but that the answer is Yes when $s\ge 6$. The case $s\in\{4,5\}$ remains open. We also prove an analogous result for planar graphs, with a necessary and sufficient hypothesis. Fix $s\ge 40$. Let $G$ be a planar graph, and let $w_0,w_1$ be its $2$ vertices with largest degrees. If there exist disjoint independent sets $I_0, I_1$ such that $|I_0|=\lfloor{n/s}\rfloor$ and $|I_1| = \lfloor{(n+1)/s}\rfloor$ and $w_0,w_1\in I_0\cup I_1$, then $G$ has an equitable $s$-coloring. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16123 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equitably Coloring Planar and Outerplanar Graphs Cranston, Daniel W. Mahmoud, Reem Combinatorics 05C15 A proper $s$-coloring of an $n$-vertex graph is \emph{equitable} if every color class has size $\lfloor{n/s}\rfloor$ or $\lceil{n/s}\rceil$. A necessary condition to have an equitable $s$-coloring is that every vertex $v$ appears in an independent set of size at least $\lfloor{n/s}\rfloor$. That is $\min_{v\in V(G)}α_v\ge \lfloor{n/s}\rfloor$. Various authors showed that when $G$ is a tree and $s\ge 3$ this obvious necessary condition is also sufficient. Kierstead, Kostochka, and Xiang asked whether this result holds more generally for all outerplanar graphs. We show that the answer is No when $s=3$, but that the answer is Yes when $s\ge 6$. The case $s\in\{4,5\}$ remains open. We also prove an analogous result for planar graphs, with a necessary and sufficient hypothesis. Fix $s\ge 40$. Let $G$ be a planar graph, and let $w_0,w_1$ be its $2$ vertices with largest degrees. If there exist disjoint independent sets $I_0, I_1$ such that $|I_0|=\lfloor{n/s}\rfloor$ and $|I_1| = \lfloor{(n+1)/s}\rfloor$ and $w_0,w_1\in I_0\cup I_1$, then $G$ has an equitable $s$-coloring. |
| title | Equitably Coloring Planar and Outerplanar Graphs |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2509.16123 |