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arXiv · 2604.02646

On the number of 4-contractible edges in plane triangulations

Toshiki Abe·Michitaka Furuya·Raiji Mukae·Shoichi Tsuchiya
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Abstract

In 2007, Ando and Egawa proved a theorem which provides a lower bound on the number of contractible edges preserving $4$-connectedness in $4$-connected graphs. In this paper, we refine their bounds, especially for the $4$-connected plane triangulations. In particular, we show that if $G$ is a $4$-connected plane triangulation of order at least $7$, then $G$ contains at least $|V_{\ge 5}|+2$ contractible edges preserving $4$-connectedness, where $V_{\ge 5}$ is the set of vertices of degree at least $5$. We also determine the extremal graphs.

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Toshiki Abe, Michitaka Furuya, Raiji Mukae, Shoichi Tsuchiya. 2026-04-03. On the number of 4-contractible edges in plane triangulations. https://arxiv.org/abs/2604.02646

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