On the balanced case of the Brualdi-Shen conjecture on 4-cycle decompositions of Eulerian bipartite tournaments

The Brualdi-Shen Conjecture on Eulerian Bipartite Tournaments states that any such graph can be decomposed into oriented 4-cycles. In this article we prove the balanced case of the mentioned conjecture. We show that for any 2n×2n bipartite graph G = (U ∪V,E) in which each vertex has n-neighbors with biadjacency matrixM (or its transpose), there is a particular proper edge coloring of a column permutation ofM denotedM. This coloring has the property that the nonzero entries at each of the first n columns are colored with elements from the set {n + 1, n + 2, . . . , 2n}, and the nonzero entries at each of the last n columns are colored with elements from the set {1, 2, . . . , n}. Moreover, if the nonzero entry M r,j is colored with color i then M r,i must be a zero entry. Such a coloring will induce an oriented 4-cycle decomposition of the bipartite tournament corresponding to M . We achieve this by constructing an euler tour on the bipartite tournament that avoids traversing both pair of edges of any two internally disjoint s-t 2-paths consecutively, where s and t belong to V .