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On restricted edge-connectivity of half-transitive multigraphs

Let $G=(V,E)$ be a multigraph (it has multiple edges, but no loops). The edge connectivity, denoted by $λ(G)$, is the cardinality of a minimum edge-cut of $G$. We call $G$ maximally edge-connected if $λ(G)=δ(G)$, and $G$ super edge-connected if every minimum edge-cut is a set of edges incident with some vertex. The restricted edge-connectivity $λ'(G)$ of $G$ is the minimum number of edges whose removal disconnects $G$ into non-trivial components. If $λ'(G)$ achieves the upper bound of restricted edge-connectivity, then $G$ is said to be $λ'$-optimal. A bipartite multigraph is said to be half-transitive if its automorphism group is transitive on the sets of its bipartition. In this paper, we will characterize maximally edge-connected half-transitive multigraphs, super edge-connected half-transitive multigraphs, and $λ'$-optimal half-transitive multigraphs.