×

zbMATH — the first resource for mathematics

Decomposition of bipartite graphs into closed trails. (English) Zbl 1224.05402
Summary: Let \(\operatorname {Lct}(G)\) 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 \(\operatorname {Lct}(G)\) 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 \(a\geq 1\) 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.

MSC:
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] P. N. Balister: Packing circuits into KN. Comb. Probab. Comput. 6 (2001), 463–499.
[2] P. N. Balister: Packing closed trails into dense graphs. J. Comb. Theory, Ser. B 88 (2003), 107–118. · Zbl 1045.05074 · doi:10.1016/S0095-8956(02)00039-4
[3] P. N. Balister: Packing digraphs with directed closed trails. Comb. Probab. Comput. 12 (2003), 1–15. · Zbl 1015.05072 · doi:10.1017/S0963548302005461
[4] Ch.-Ch. Chou, Ch.-M. Fu and W.-Ch. Huang: Decomposition of K n,m into short cycle. Discrete Math. 197/198 (1999), 195–203.
[5] S. Cichacz: Decomposition of complete bipartite digraphs and even complete bipartite multigraphs into closed trails. Discuss. Math. Graph Theory 27 (2007), 241–249. · Zbl 1133.05075
[6] S. Cichacz, J. Przybyło and M. Woźniak: Decompositions of pseudographs into closed trails of even sizes. Discrete Math., doi:10.1016/j.disc.2008.04.024. · Zbl 1218.05123
[7] M. Horňák and Z. Kocková: On complete tripartite graphs arbitrarily decomposable into closed trails. Tatra Mt. Math. Publ. 36 (2007), 71–107. · Zbl 1174.05073
[8] M. Horňák and M. Woźniak: Decomposition of complete bipartite even graphs into closed trails. Czech. Math. J. 53 (2003), 127–134. · Zbl 1010.05054 · doi:10.1023/A:1022931710349
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.