On the number of edge-colourings of regular bipartite graphs

Abstract

Let f( k) be the largest number such that each k-regular bipartite graph with 2 n vertices has at least ƒ(k) n 1-factorizations. We prove that ƒ(k)⩽ k! 2 k k , and that equality holds if k contains no other prime factors than 2 and 3. We conjecture equality for each k.