| Throughout this paper,a graph is assumed to be finite,simple and undirected.For a graph X,every edge of X gives rise to a pair of opposite arcs.By V(X),E(X),A(X)and Aut(X),we denote the vertex set,edge set,arc set and the full automorphism group of a graph X respectively.Use Kn,n-nK2 to represent the complete bipartite graph of n points minus a perfect match.A 2-arc of X is a sequence(v0,v1,v2)of three vertices such that(v0,v1),(v1,v2)in A(Y)but v0≠v2.The graph X is said to be 2-arc-transitive if Aut(X)acts transitively on the set of 2-arcs of X.A graph X is called a covering of a graph Y and p is called a covering projection from X to Y,if there is a surjection p:V(X)→V(Y)such that P|N(x):N(x)→N(y)is a bijection for any vertex y∈V(Y)and x∈p-1(y).The graph X is called the covering graph and Y is the base graph.The fiber of an edge or a vertex is its preimage under p.An automorphism of X which maps a fiber to a fiber is said to be fiber-preserving.All such groups composed of fiber preserving automorphisms are called fiber preserving automorphism groups.The group K of all automorphisms of X which fix each of the fibers setwise is called the covering transformation group.It is easy to get that if X is a connected graph,the role of K on each point fiber is semi regular.In particular,if the action is free and transitive(i.e.regularly),X is said to be a regular covering of Y.In this article[J.Combin.Theory Ser.B 111(2015),54-74],the authors Classified the regular covers of complete graphs Kn,where the covering transformation group is a metacyclic group and the fiber preserving automorphism group acts 2-arc-transitively.In this paper,we continue to classify the regular covers of complete bipartite graph minus a perfect matching Kn,n-nK2 with a metacyclic covering transformation group.In the first chapter,we give the introduction;The second chapter expounds some preliminary results;In the third chapter,we discuss n≥6 and n=5 respectively;The fourth chapter gives the proof of n=4. |
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