The Existence of Completely Independent Spanning Trees for Some Compound Graphs

Given two regular graphs $G$G and $H$H such that the vertex degree of $G$G is equal to the number of vertices in $H$H , the compound graph $G(H)$G(H) is constructed by replacing each vertex of $G$G by a copy of $H$H and replacing each edge of $G$G by an additional edge connecting random vertices in two corresponding copies of $H$H , respectively, under the constraint that each vertex in $G(H)$G(H) is incident with only one additional edge, exactly. $L$L - $HSDC_m(m)$HSDCm(m) is a compound graph $G(H)$G(H) , where $G$G is a hypercube $Q_m$Qm and $H$H is a complete graph $K_m$Km , which is defined by focusing on the connected relation between servers in the novel data center network $HSDC_m(m)$HSDCm(m) proposed in [30]. A set of $k$k spanning trees in a graph $G$G are called completely independent spanning trees (CISTs for short) if the paths joining every pair of vertices $x$x and $y$y in any two trees have neither vertex nor edge in common, except for $x$x and $y$y . In this paper, we give a sufficient condition for the existence of $k$k CISTs in a kind of compound graph. Furthermore, a specific construction algorithm is provided. As corollaries of the main results, the existences of two CISTs for $m\geq 4$m≥4 ; three CISTs for $m\geq 8$m≥8 and four CISTs for $m\geq 10$m≥10 in $L$L - $HSDC_m(m)$HSDCm(m) are gotten directly.