On Perfect Matching Coverings and Even Subgraph Coverings

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 Contract grant sponsor: CNNSF; Contract grant number: 11271348; Contract grant sponsor: NSA; Contract grant numbers: H98230-12-1-0233 andH98230-14-1-0154; Contract grant sponsor: NSF; Contract grant number: DMS-1264800. Journal of Graph Theory C © 2015 Wiley Periodicals, Inc. 83 84 JOURNAL OF GRAPH THEORY a Berge covering. We also obtain some results concerning the so-called 5-even subgraph double cover conjecture. C © 2015 Wiley Periodicals, Inc. J. Graph Theory 81: 83–91, 2016