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arXiv · 1903.08795

Largest 2-regular subgraphs in 3-regular graphs

Ilkyoo Choi·Ringi Kim·Alexandr Kostochka·Boram Park·Douglas B. West
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Abstract

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph with exactly $c$ cut-edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (c-1)/2\rfloor\}$ vertices. More generally, every $n$-vertex multigraph with maximum degree $3$ and $m$ edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (3n-2m+c-1)/2\rfloor\}$ vertices. These bounds are sharp; we describe the extremal multigraphs.

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Ilkyoo Choi, Ringi Kim, Alexandr Kostochka, Boram Park, Douglas B. West. 2019-03-21. Largest 2-regular subgraphs in 3-regular graphs. https://arxiv.org/abs/1903.08795

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