Atoms of cyclic edge connectivity in regular graphs

Abstract

A cyclic edge-cut of a connected graph $G$ is an edge set, the removal of which separates two cycles. If $G$ has a cyclic edge-cut, then it is called cyclically separable. For a cyclically separable graph $G$, the cyclic edge connectivity of a graph $G$, denoted by $\lambda _c(G)$, is the minimum cardinality over all cyclic edge cuts. Let $X$ be a non-empty proper subset of $V(G)$. If $[X,\overline{X}]=\{xy\in E(G)\ |\ x\in X, y\in \overline{X}\}$ is a minimum cyclic edge cut of $G$, then $X$ is called a $\lambda _c$ -fragment of $G$. A $\lambda _c$-fragment with minimum cardinality is called a $\lambda _c$ -atom. Let $G$ be a $k (k\ge 3)$-regular cyclically separable graph with $\lambda _c(G)<g(k-2)$, where $g$ is the girth of $G$. A combination of the results of Nedela and Skoviera (Math Slovaca 45:481–499, 1995) and Xu and Liu (Australas J Combin 30:41–49, 2004) gives that if $k\ne 5$ then any two distinct $\lambda _c$-atoms of $G$ are disjoint. The remaining case of $k=5$ is considered in this paper, and a new proof for Nedela and Škoviera’s result is also given. As a result, we obtain the following result. If $X$ and $X'$ are two distinct $\lambda _c$-atoms of $G$ such that $X\cap X'\ne \emptyset$, then $(k,g)=(5,3)$ and $G[X]\cong K_4$. As corollaries, several previous results are easily obtained.