Mimimal graphs for completely independent spanning trees and completely independent spanning trees in complete t-partite graph
Abstract
Let $T_{1},T_{2},\dots,T_{k}$ be spanning trees of a graph $G$. For any two vertices
$u,v$ of $G$, if the paths from $u$ to $v$ in these $k$ trees are pairwise openly disjoint, then we say that $T_{1},T_{2},\dots,T_{k}$ are completely independent spanning trees. In this paper, we give the definition of Minimal graph for $k$ completely independent spanning trees and we characterized all Minimal graphs for $k$ completely independent spanning trees. Finally, we obtain the number of completely independent spanning trees in complete $t(t\geq 2)$-partite graph $K_{n_{1},n_{2},\cdots,n_{t}}$, which is generalize the known result.
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