Completely Independent Spanning Trees in Some Regular Graphs

Benoit Darties (1) , Nicolas Gastineau (2) , Olivier Togni (2)

1 Le2i - Laboratoire Electronique, Informatique et Image [UMR6306]
2 Equipe Combinatoire

Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices, except the vertices $u$ and $v$, then $T_1,\ldots, T_k$ are completely independent spanning trees in $G$. For $2k$-regular graphs which are $2k$-connected, such as the Cartesian product of a complete graph of order $2k-1$ and a cycle and some Cartesian products of three cycles (for $k=3$), the maximum number of completely independent spanning trees contained in these graphs is determined and it turns out that this maximum is not always $k$.

Mots clés: Spanning tree; Cartesian product; Completely independent spanning tree.; Completely independent spanning tree

Domaines: Mathématique discrète [cs.DM]

Benoit Darties, Nicolas Gastineau, Olivier Togni. Completely Independent Spanning Trees in Some Regular Graphs. 2014. ⟨hal-01066448⟩