A nearly equitable edge-coloring of a multigraph is a coloring such that edges incident to each vertex are colored equitably in number. This problem was solved in O(kn2) time, where n and k are the numbers of the edges and the colors, respectively. The running time was improved to be O(n2/k + n|V|) later. We present a more efficient algorithm for this problem that runs in O(n2/k) time.

Karger, Motwani and Sudan presented a graph coloring algorithm based on semidefinite programming, which colors any k-colorable graph with maximum degree Δ using (Δ1-2/k) colors. This algorithm leads to an algorithm for k-colorable graph using (n 1-3/(k+1)) colors. This improved Wigderson's algorithm, which uses O(n1-1/(k-1)) colors, containing as a subroutine an algorithm using (Δ+1) colors for graphs with maximum degree Δ. It is easy to imagine that an algorithm which uses less colors in terms of Δ leads to an algorithm which uses less colors in terms of n. In this paper, we consider this influence assuming that we have an algorithm which uses (Δ 1-x/k) colors for 2