On graph decompositions modulo k

We prove that, for every integer k ≥ 2, every graph has an edge-partition into 5k log k sets, each of which is the edge-set of a graph with all degrees congruent to 1 mod k. This answers a question of Pyber. Pyber [8] proved that every graph G has an edge-partition into four sets, each of which is the edge set of a graph with all degrees odd; if every component of G has even order then three sets will do. This is best possible, as can be seen by considering K5 with two independent edges removed, which cannot be partitioned into fewer than four subgraphs with all degrees odd; and K4 with one edge removed, which requires three. Motivated by this result, Pyber [8] asked what happens when we consider residues mod k, rather than mod 2. In particular he asked whether for every integer k there is an integer c(k) such that every graph has an edge-partition into at most c(k) sets, each of which is the edge-set of a