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Czechoslovak Mathematical Journal, Vol. 59, No. 1, pp. 129-144, 2009
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Decomposition of bipartite graphs into closed trails

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Sylwia Cichacz, Mirko Hornak

* Sylwia Cichacz*, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland, e-mail: ` cichacz@agh.edu.pl`; * Mirko Hornak*, P. J. Safarik University, Jesenna 5, 040 01 Kosice, Slovakia, e-mail: ` mirko.hornak@upjs.sk`

**Abstract:** Let ${\Lct}(G)$ denote the set of all lengths of closed trails that exist in an even graph $G$. A sequence $(t_1,\dots,t_p)$ of elements of ${\Lct}(G)$ adding up to $|E(G)|$ is $G$-realisable provided there is a sequence $(T_1,\dots,T_p)$ of pairwise edge-disjoint closed trails in $G$ such that $T_i$ is of length $t_i$ for $i=1,\dots,p$. The graph $G$ is arbitrarily decomposable into closed trails if all possible sequences are $G$-realisable. In the paper it is proved that if $a\ge1$ is an odd integer and $M_{a,a}$ is a perfect matching in $K_{a,a}$, then the graph $K_{a,a}-M_{a,a}$ is arbitrarily decomposable into closed trails.

**Keywords:** even graph, closed trail, graph arbitrarily decomposable into closed trails, bipartite graph

**Classification (MSC 2000):** 05C70

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