On Perfect Matching Coverings and Even Subgraph Coverings. Issue 1 (4th March 2015) - Hou, Xinmin; Lai, Hong‐Jian; Zhang, Cun‐Quan
Abstract: A perfect matching covering of a graph G is a set of perfect matchings of G such that every edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 (we call such a collection of perfect matchings a Berge covering of G ). A cubic graph G is called a Kotzig graph if G has a 3‐edge‐coloring such that each pair of colors forms a hamiltonian circuit (introduced by R. Häggkvist, K. Markström, J Combin Theory Ser B 96 (2006), 183–206). In this article, we prove that if there is a vertex w of a cubic graph G such that G − w ¯, the graph obtained from G − w by suppressing all degree two vertices is a Kotzig graph, then G has a Berge covering. We also obtain some results concerning the so‐called 5‐even subgraph double cover conjecture.