The Existence of Completely Independent Spanning Trees for Some Compound Graphs

Abstract

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 - $HSDC_m(m)$ H S D C m ( m ) is a compound graph $G(H)$ G ( H ) , where $G$ G is a hypercube $Q_m$ Q m and $H$ H is a complete graph $K_m$ K m , which is defined by focusing on the connected relation between servers in the novel data center network $HSDC_m(m)$ H S D C m ( 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 - $HSDC_m(m)$ H S D C m ( m ) are gotten directly.