Cliques in Squares of Graphs with Maximum Average Degree less than 4
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929615826583552 |
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| author | Cranston, Daniel W. Yu, Gexin |
| author_facet | Cranston, Daniel W. Yu, Gexin |
| contents | Hocquard, Kim, and Pierron constructed, for every even integer $D\ge 2$, a 2-degenerate graph $G_D$ with maximum degree $D$ such that $ω(G_D^2)=\frac52D$. We prove for (a) all 2-degenerate graphs $G$ and (b) all graphs $G$ with $\mbox{mad}(G)<4$, upper bounds on the clique number $ω(G^2)$ of $G^2$ that match the lower bound given by this construction, up to small additive constants. We show that if $G$ is 2-degenerate with maximum degree $D$, then $ω(G^2)\le \frac52D+72$ (with $ω(G^2)\le \frac52D+60$ when $D$ is sufficiently large). And if $G$ has $\mbox{mad}(G)<4$ and maximum degree $D$, then $ω(G^2)\le \frac52D+532$. Thus, the construction of Hocquard et al. is essentially best possible. Our proofs introduce a "token passing" technique to derive crucial information about non-adjacencies in $G$ of vertices that are adjacent in $G^2$. This is a powerful technique for working with such graphs that has not previously appeared in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11763 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cliques in Squares of Graphs with Maximum Average Degree less than 4 Cranston, Daniel W. Yu, Gexin Combinatorics 05C69, 05C35, 05C15 Hocquard, Kim, and Pierron constructed, for every even integer $D\ge 2$, a 2-degenerate graph $G_D$ with maximum degree $D$ such that $ω(G_D^2)=\frac52D$. We prove for (a) all 2-degenerate graphs $G$ and (b) all graphs $G$ with $\mbox{mad}(G)<4$, upper bounds on the clique number $ω(G^2)$ of $G^2$ that match the lower bound given by this construction, up to small additive constants. We show that if $G$ is 2-degenerate with maximum degree $D$, then $ω(G^2)\le \frac52D+72$ (with $ω(G^2)\le \frac52D+60$ when $D$ is sufficiently large). And if $G$ has $\mbox{mad}(G)<4$ and maximum degree $D$, then $ω(G^2)\le \frac52D+532$. Thus, the construction of Hocquard et al. is essentially best possible. Our proofs introduce a "token passing" technique to derive crucial information about non-adjacencies in $G$ of vertices that are adjacent in $G^2$. This is a powerful technique for working with such graphs that has not previously appeared in the literature. |
| title | Cliques in Squares of Graphs with Maximum Average Degree less than 4 |
| topic | Combinatorics 05C69, 05C35, 05C15 |
| url | https://arxiv.org/abs/2305.11763 |