Induced Cycles in Graphs

TL;DR

This paper establishes lower bounds on the size of maximum induced 2-regular subgraphs in regular and claw-free graphs, providing new theoretical insights into their structure and extremal properties.

Contribution

It proves new bounds for induced 2-regular subgraphs in regular and cubic claw-free graphs, advancing understanding of their extremal characteristics.

Findings

For r-regular graphs, c_ind(G) ≥ n/(2(r-1)) + 1/((r-1)(r-2))

In cubic claw-free graphs, c_ind(G) > 13n/20

Bounds are tight or asymptotically optimal

Abstract

The maximum cardinality of an induced $2$-regular subgraph of a graph $G$ is denoted by $c_{\rm ind}(G)$. We prove that if $G$ is an $r$-regular graph of order $n$, then $c_{\rm ind}(G) \geq \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}$ and we prove that if $G$ is a cubic claw-free graph on order $n$, then $c_{\rm ind}(G) > 13n/20$ and this bound is asymptotically best possible.