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Bipartite graphs decomposable into closed trails. (English) Zbl 1223.05225
Summary: 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 ([S. Cichacz and M. Horňák, Czech. Math. J. 59, No. 1, 129–144 (2009; Zbl 1224.05402); M. Horňák and M. Woźniak, Czech. Math. J. 53, No. 1, 127–134 (2003)]) 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 ([E. J. Billington, Matematiche 59, No. 1–2, 53–72 (2004; Zbl 1195.05058)]).