Triangle-free Graphs with Large Minimum Common Degree
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911983818768384 |
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| author | Wang, Jian Yang, Weihua Zhao, Fan |
| author_facet | Wang, Jian Yang, Weihua Zhao, Fan |
| contents | Let $G$ be a graph. For $x\in V(G)$, let $N(x)=\{y\in V(G)\colon xy\in E(G)\}$. The minimum common degree of $G$, denoted by $δ_{2}(G)$, is defined as the minimum of $|N(x)\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982, Häggkvist showed that every triangle-free graph with minimum degree greater than $\lfloor\frac{3n}{8}\rfloor$ is homomorphic to a cycle of length 5. In this paper, we prove that every triangle-free graph with minimum common degree greater than $\lfloor\frac{n}{8}\rfloor$ is homomorphic to a cycle of length 5, which implies Häggkvist's result. The balanced blow-up of the Möbius ladder graph shows that it is best possible. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_05547 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Triangle-free Graphs with Large Minimum Common Degree Wang, Jian Yang, Weihua Zhao, Fan Combinatorics Let $G$ be a graph. For $x\in V(G)$, let $N(x)=\{y\in V(G)\colon xy\in E(G)\}$. The minimum common degree of $G$, denoted by $δ_{2}(G)$, is defined as the minimum of $|N(x)\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982, Häggkvist showed that every triangle-free graph with minimum degree greater than $\lfloor\frac{3n}{8}\rfloor$ is homomorphic to a cycle of length 5. In this paper, we prove that every triangle-free graph with minimum common degree greater than $\lfloor\frac{n}{8}\rfloor$ is homomorphic to a cycle of length 5, which implies Häggkvist's result. The balanced blow-up of the Möbius ladder graph shows that it is best possible. |
| title | Triangle-free Graphs with Large Minimum Common Degree |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.05547 |