Double cycle covers and the petersen graph
Paul A. Catlin · Journal of Graph Theory · 1989
Abstract Let O(G) denote the set of odd‐degree vertices of a graph G. Let t ϵ N and let 𝒮t denote the family of graphs G whose edge set has a partition. E(g) = E1 U E2 U … U Etsuch that O(G) = O(G[Ei]) (1 ⩽ i ⩽ t). This partition is associated with a double cycle cover of G. We show that if a graph G is at most 5 edges short of being 4‐edge‐connected, then exactly one of these holds: G ϵ 𝒮3, G has at least one cut‐edge, or G is contractible to the Petersen graph. We also improve a sufficient condition of Jaeger for G ϵ 𝒮2p+1(p ϵ N).