An algorithm to construct independent spanning trees on parity cubes

Abstract

Independent spanning trees have applications in networks such as reliable communication protocols, one-to-all broadcasting, reliable broadcasting, and secure message distribution. Thus, the designs of independent spanning trees in several classes of networks have been widely investigated. However, there is a conjecture on independent spanning trees: any n-connected graph has n independent spanning trees rooted at an arbitrary vertex. This conjecture still remains open for n≥5. In this paper, by proposing an algorithm to construct n independent spanning trees rooted at any vertex, we confirm the conjecture on n-dimensional parity cube PQn —— a variant of n-dimensional hypercube. Furthermore, we prove that all independent spanning trees rooted at an arbitrary vertex constructed by our construction method are isomorphic and the height of each tree is n+1 for any integer n≥2.