$c\lambda$-Optimally Connected Mixed Cayley Graph

Abstract

The cyclic edge-connectivity of a cyclically separable graph $G$, denoted by $c\lambda (G)$, is the minimum cardinality of all edge subsets $F$ such that $G–F$ is disconnected and at least two of its components contain cycles. Since $c\lambda (G)\le \zeta (G)$, where $\zeta (G)=min\{w(A)\mid A\text{ induces a shortest cycle in }G\}$, for any cyclically separable graph $G$, a cyclically separable graph $G$ is said to be cyclically optimal if $c\lambda (G)=\zeta (G)$. The mixed Cayley graph is a kind of semi-regular graph. The cyclic edge-connectivity is a widely studied parameter, which can be used to measure the reliability of a network. Because previous work studied cyclically optimal mixed Cayley graphs with girth $g\ge 5$, this paper focuses on mixed Cayley graphs with girth $g<5$ and gives some sufficient and necessary conditions for these graphs to be cyclically optimal.