Bipartite Graphs Decomposable into Closed Trails

Let \( G = K_{a,b} \), where \( a, b \) are even, or \( G = K_{a,a} – M_{2a} \), where \( a \geq 1 \) is an odd integer and \( M_{2a} \) is a perfect matching in \( K_{a,a} \). It has been shown ([3,4]) that \( G \) is arbitrarily decomposable into closed trails. Billington asked if the graph \( K_{r,s} – F \), where \( s, r \) are odd and \( F \) is a (smallest possible) spanning subgraph of odd degree, is arbitrarily decomposable into closed trails ([2]).

In this article we answer the question in the affirmative.