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 bedecomposed into oriented 4-cycles. In this article we prove the balanced case of the mentionedconjecture. We show that for any 2nx2n bipartite graph G = (U(cup)V;E) in which each vertex hasn-neighbors with biadjacency matrixM (or its transpose), there is a particular proper edge coloringof a column permutation of M denotedMσ. This coloring has the property that the nonzero entriesat 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 Mσ r;j is colored with color i then Mσ r;i must be a zero entry. Such a coloring will induce an oriented 4-cycle decomposition of the bipartite tournament corresponding toM. 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 .