New Bounds for the Nearly Equitable Edge Coloring Problem

Abstract

An edge coloring of a multigraph is nearly equitable if, among the edges incident to each vertex, the numbers of edges colored with any two colors differ by at most two. It has been proved that this problem can be solved in O(m ²/k) time, where m and k are the numbers of edges and given colors, respectively. In this paper, we present a recursive algorithm that runs in $O\left(mn \log\left(m/(kn) +1\right) \right)$ time, where n is the number of vertices. This algorithm improves the best-known worst-case time complexity. When k = O(1), the time complexity of all known algorithms is O(m ²), which implies that this time complexity remains to be the best for more than twenty years since 1982 when Hilton and de Werra gave a constructive proof for the existence of a nearly equitable edge coloring for any graph. Our result is the first that improves this time complexity when m/n grows to infinity; e.g., m = n ^ϑ for an arbitrary constant ϑ> 1. We also propose a very simple randomized algorithm that runs in $O\left(m^{3/2}n^{1/2}/k^{1/2}\right)$ time with probability at least 1 − 1/c for any constant c > 1, whose worst-case time complexity is O(m ²/k).