On the enumeration of bipartite minimum edge colorings

For a bipartite graph G = (V., E), an edge coloring of G is a coloring of the edges of G such that, any two adjacent edges are colored in different colors. In this paper, we consider the problem of enumerating all edge colorings with the fewest number of colors. We propose a simple polyuomial delay algorithm whose amortized time complexity is O(vertical bar V vertical bar) per output, whereas the previous fastest algorithm took O(vertical bar E vertical bar log vertical bar V vertical bar) time per output.. Although the delay of the algorithm is O(vertical bar E vertical bar vertical bar V vertical bar), the delay of our algorithm can be reduced to O(vertical bar V vertical bar) by using a simple modification with a queue of polynomial size. We show an improvement to reduce the space complexity from 0(vertical bar V vertical bar vertical bar E vertical bar) to O(vertical bar E vertical bar + vertical bar V vertical bar). Furthermore, we obtain a lower bound (' E vertical bar - vertical bar(V) over cap vertical bar) max {2(Delta-3), 2(vertical bar(V) over cap vertical bar/2 + 1)(Delta-3)/(Delta - 1)}/Delta of the number of edge colorings included in G, where A is the maximum degree and 1 is the set of vertices of the maximum degree.