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arXiv · 2211.07913

Extremal graphs for the suspension of edge-critical graphs

Jianfeng Hou·Heng Li·Qinghou Zeng
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Abstract

The Turán number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\mathcal{F}$ of graphs, the suspension $\mathcal{F}+v$ of $\mathcal{F}$ is the graph obtained by connecting the vertex $v$ to all vertices of $F$ for each $F\in \mathcal{F}$. For two integers $k\ge1$ and $r\ge2$, let $H_i$ be a graph containing a critical edge with chromatic number $r$ for any $i\in\{1,\ldots,k\}$, and let $H=\{H_1,\ldots,H_k\}+v$. In this paper, we determine $\text{ex}(n, H)$ and characterize all the extremal graphs for sufficiently large $n$. This generalizes a result of Chen, Gould, Pfender and Wei on intersecting cliques. We also obtain a stability theorem for $H$, extending a result of Roberts and Scott on graphs containing a critical edge.

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Jianfeng Hou, Heng Li, Qinghou Zeng. 2022-11-15. Extremal graphs for the suspension of edge-critical graphs. https://arxiv.org/abs/2211.07913

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