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arXiv · 2410.23702

The minimum size of a $3$-connected locally nonforesty graph

Chengli Li·Yurui Tang·Xingzhi Zhan
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Abstract

A local subgraph of a graph is the subgraph induced by the neighborhood of a vertex. Thus a graph of order $n$ has $n$ local subgraphs. A graph $G$ is called locally nonforesty if every local subgraph of $G$ contains a cycle. Recently, in studying forest cuts of a graph, Chernyshev, Rauch and Rautenbach posed the conjecture that if $n$ and $m$ are the order and size of a $3$-connected locally nonforesty graph respectively, then $m\ge 7(n-1)/3.$ We solve this problem by determining the minimum size of a $3$-connected locally nonforesty graph of order $n.$ It turns out that the conjecture does not hold.

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Chengli Li, Yurui Tang, Xingzhi Zhan. 2024-10-31. The minimum size of a $3$-connected locally nonforesty graph. https://arxiv.org/abs/2410.23702

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