Circuit [formula omitted]-covers of signed graphs

Abstract

Let G be a signed graph and F a set of signed circuits in G. For an edge e∈E(G), F(e) denotes the number of signed circuits in F that contain e. F is called a circuit-cover of G if F(e)>0 for each e∈E(G), and a circuit k-cover of G if F(e)=k for each e∈E(G). G is coverable if it has a circuit-cover. The existence of a circuit-cover in G is equivalent to the existence of a nowhere-zero flow in G. For a coverable signed graph G, it is proved in this paper that if each maximal 2-edge-connected subgraph of G is eulerian, then G has a circuit 6-cover, consisting of four circuit-covers of G, and as an immediate consequence, G has a circuit-cover of length at most 32|E(G)|, generalizing a known result on signed eulerian graphs. New results on circuit k-covers are obtained and applied to estimating bounds on the lengths of shortest circuit-covers of signed graphs.