Factorizations of regular graphs

AbstractLet G be a k-regular graph of order 2n such that k≥n. Hilton (J. Graph Theory, 9 (1985), 193–196) proved that G contains at least ⌊k3⌋ edge-disjoint 1-factors. Hilton's theorem is improved in this paper that G contains at least ⌊k2⌋ edge-disjoint 1-factors. The following result is also proved in this paper: Let G be a 2-connected, k-regular, non-bipartite graph of order at most 3k − 3 and x, y be a pair of distinct vertices. If Gβ {x, y} is connected, then G contains an (x, y)-Hamilton path