Space-Efficient Euler Partition and Bipartite Edge Coloring

We describe space-efficient algorithms for two problems on undirected multigraphs: Euler partition (partitioning the edges into a minimum number of trails); and bipartite edge coloring (coloring the edges of a bipartite multigraph with the minimum number of colors). Let n, m and Delta >= 1 be the numbers of vertices and of edges and the maximum degree, respectively, of the input multigraph. For Euler partition we reduce the amount of working memory needed by a logarithmic factor, to O(n+m) bits, while preserving a running time of O(n+m). For bipartite edge coloring, still using O(n+m) bits of working memory, we achieve a running time of O(n+m min{Delta, log Delta(log* Delta+(log m log Delta)/Delta)}). This is O(m log Delta log* Delta) if m = Omega(n log n log log n/log* n), to be compared with O(m log Delta) for the fastest known algorithm.