# Article

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Keywords:
even graph; closed trail; graph arbitrarily decomposable into closed trails; bipartite graph
Summary:
Let ${\rm Lct}(G)$ 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 ${\rm Lct}(G)$ 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 $a\ge 1$ 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.
References:
[1] Balister, P. N.: Packing circuits into $K_N$. Comb. Probab. Comput. 6 (2001), 463-499. MR 1869841 | Zbl 1113.05309
[2] Balister, P. N.: Packing closed trails into dense graphs. J. Comb. Theory, Ser. B 88 (2003), 107-118. DOI 10.1016/S0095-8956(02)00039-4 | MR 1973263 | Zbl 1045.05074
[3] Balister, P. N.: Packing digraphs with directed closed trails. Comb. Probab. Comput. 12 (2003), 1-15. DOI 10.1017/S0963548302005461 | MR 1967482 | Zbl 1015.05072
[4] Chou, Ch.-Ch., Fu, Ch.-M., Huang, W.-Ch.: Decomposition of $K_{n,m}$ into short cycle. Discrete Math. 197/198 (1999), 195-203. MR 1674862
[5] Cichacz, S.: Decomposition of complete bipartite digraphs and even complete bipartite multigraphs into closed trails. Discuss. Math. Graph Theory 27 (2007), 241-249. DOI 10.7151/dmgt.1358 | MR 2355718 | Zbl 1133.05075
[6] Cichacz, S., Przybyło, J., Wo'zniak, M.: Decompositions of pseudographs into closed trails of even sizes. Discrete Math., doi:10.1016/j.disc.2008.04.024. DOI 10.1016/j.disc.2008.04.024
[7] Horňák, M., Kocková, Z.: On complete tripartite graphs arbitrarily decomposable into closed trails. Tatra Mt. Math. Publ. 36 (2007), 71-107. MR 2378742
[8] Horňák, M., Wo'zniak, M.: Decomposition of complete bipartite even graphs into closed trails. Czech. Math. J. 53 (2003), 127-134. DOI 10.1023/A:1022931710349 | MR 1962004

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