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Abstract: Let $\mathrm{Lct}(G)$ denote the set of all lengths of closed trails that exist in an even graph $G$. A sequence $(t_1,\dots,t_p)$ of terms of $\mathrm{Lct}(G)$ adding up to $|E(G)|$ is $G$-realizable 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 $G$-realizable. 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. Contact the authors: cichacz@agh.edu.pl, mirko.hornak@upjs.sk Download PDF version of the preprint. |