Fulkerson′s Conjecture and Circuit Covers

1‐Factor and Cycle Covers of Cubic Graphs

“…Let G c be a bridgeless k-core of G. By Theorem 2.16, G c has a cycle cover of length at most 5 3 |E(G c )|. By Lemma 2.4 we have |E(G c )| = 2k − |E 3 |, and hence, it follows with Theorem 2.15 that G has a cycle cover of length at most 4 3 |E(G)| + 2k. We are going to prove better bounds for the length of cycle covers of a cubic graphs which have a bipartite core.…”

“…Thus, k < 1 3 |E(G)| and Theorem 2.19 implies that G has an even 3-cycle cover of length smaller than 14 9 |E(G)|. In [4] it is proved that if a cubic graph G has a Fulkerson-cover, then it has a 3-cycle cover of length at most 22 15 |E(G)|. This bound is best possible for 3-cycle covers of bridgeless cubic graphs, since it is attained by the Petersen graph [11].…”

“…Thus, there are two of these three cycles covering E 1 ∪E 2 with total length at most 4 3 |E 1 ∪E 2 | ≤ 4 3 (|E(G)|−k). These two cycles together with a cycle cover of G c gives a cycle cover of G with length at most 4 3 (|E(G)| − k) + t. The cycle cover is even if and only if the cycle cover of G c is even.…”

“…Hence, φ induces a proper 3-edge-coloring φ ′ on H c , where E * is a color class. Let C Hc be a cycle cover of H c of length 4 3 |E(H c )|, which is induced by φ ′ and which uses the edges of E * twice. Now, C Hc induces a 2-cycle cover C Kc of K c .…”

“…Corollary 2.20 Let G be a triangle-free bridgeless cubic graph. If µ 3 (G) ≤ 5, then G has an even 3-cycle cover of length at most 4 3 |E(G)| + 2.…”