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arXiv · 2102.01338

On the minimal degree condition of graphs implying some properties of subgraphs

Bingchen Qian·Chengfei Xie·Gennian Ge
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Abstract

Erd\H{o}s posed the problem of finding conditions on a graph $G$ that imply the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We generalize this problem to general cases. Let $\delta_r$ be the least number so that any graph $G$ on $n$ vertices with minimum degree $\delta_rn$ has the property $P_{r-1}(G)=K_rf(G),$ where $P_{r-1}(G)$ is the largest number of edges in an $(r-1)$-partite subgraph and $K_rf(G)$ is the largest number of edges in a $K_r$-free subgraph. We show that $\frac{3r-4}{3r-1}<\delta_r\le\frac{4(3r-7)(r-1)+1}{4(r-2)(3r-4)}$ when $r\ge4.$ In particular, $\delta_4\le 0.9415.$

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Bingchen Qian, Chengfei Xie, Gennian Ge. 2021-02-02. On the minimal degree condition of graphs implying some properties of subgraphs. https://arxiv.org/abs/2102.01338

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