Bounding Clique Size in Squares of Planar Graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913598207426560 |
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| author | Cranston, Daniel W. |
| author_facet | Cranston, Daniel W. |
| contents | Wegner conjectured that if $G$ is a planar graph with maximum degree $Δ\ge 8$, then $χ(G^2)\le \left\lfloor \frac32Δ\right\rfloor +1$. This problem has received much attention, but remains open for all $Δ\ge 8$. Here we prove an analogous bound on $ω(G^2)$: If $G$ is a plane graph with $Δ(G)\ge 36$, then $ω(G^2)\le \lfloor\frac32Δ(G)\rfloor+1$. In fact, this is a corollary of the following lemma, which is our main result. If $G$ is a plane graph with $Δ(G)\ge 19$ and $S$ is a maximal clique in $G^2$ with $|S|\ge Δ(G)+20$, then there exist $x,y,z\in V(G)$ such that $S=\{w:|N[w]\cap\{x,y,z\}|\ge 2\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_09585 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounding Clique Size in Squares of Planar Graphs Cranston, Daniel W. Combinatorics 05C15, 05C10, 05C69, 05C12 Wegner conjectured that if $G$ is a planar graph with maximum degree $Δ\ge 8$, then $χ(G^2)\le \left\lfloor \frac32Δ\right\rfloor +1$. This problem has received much attention, but remains open for all $Δ\ge 8$. Here we prove an analogous bound on $ω(G^2)$: If $G$ is a plane graph with $Δ(G)\ge 36$, then $ω(G^2)\le \lfloor\frac32Δ(G)\rfloor+1$. In fact, this is a corollary of the following lemma, which is our main result. If $G$ is a plane graph with $Δ(G)\ge 19$ and $S$ is a maximal clique in $G^2$ with $|S|\ge Δ(G)+20$, then there exist $x,y,z\in V(G)$ such that $S=\{w:|N[w]\cap\{x,y,z\}|\ge 2\}$. |
| title | Bounding Clique Size in Squares of Planar Graphs |
| topic | Combinatorics 05C15, 05C10, 05C69, 05C12 |
| url | https://arxiv.org/abs/2308.09585 |