Decomposition of bipartite graphs into closed trails

Cichacz, Sylwia ; Horňák, Mirko

Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 129-144.

Résumé

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.

Détail

Classification : 05C38, 05C70
Keywords: even graph; closed trail; graph arbitrarily decomposable into closed trails; bipartite graph