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arXiv · 2411.19773

Complete tripartite subgraphs of balanced tripartite graphs with large minimum degree

Yihan Chen·Jialin He·Allan Lo·Cong Luo·Jie Ma·Yi Zhao
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Abstract

In 1975 Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di asked what minimum degree guarantees an octahedral subgraph $K_3(2)$ in any tripartite graph $G$ with $n$ vertices in each vertex class. We show that $\delta(G)\geq n+2n^{\frac{5}{6}}$ suffices thus improving the bound $n+(1+o(1))n^{\frac{11}{12}}$ of Bhalkikar and Zhao obtained by following their approach. Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di conjectured that $n+cn^{\frac{1}{2}}$ suffices and there are many $K_3(2)$-free tripartite graphs $G$ with $\delta(G)\geq n+cn^{\frac{1}{2}}$. We confirm this conjecture under the additional assumption that every vertex in $G$ is adjacent to at least $(1/5+\varepsilon)n$ vertices in any other vertex class.

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Yihan Chen, Jialin He, Allan Lo, Cong Luo, Jie Ma, Yi Zhao. 2024-11-29. Complete tripartite subgraphs of balanced tripartite graphs with large minimum degree. https://arxiv.org/abs/2411.19773

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