Space-Efficient Euler Partition and Bipartite Edge Coloring

Abstract

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 $\varDelta \ge 1$ 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 $O(n+m\min \{\varDelta ,\log \varDelta ( \log ^*\!\varDelta + (\log m \log \varDelta ) /\varDelta )\})$. This is $O(m \log \varDelta \log ^*\!\varDelta )$ if $m = \varOmega ({{n \log n \log \log n}/{\log ^*\!n}})$, to be compared with $O(m\log \varDelta )$ for the fastest known algorithm.