5 research outputs found
Geodesic bipancyclicity of the Cartesian product of graphs
A cycle containing a shortest path between two vertices and in a graph is called a -geodesic cycle. A connected graph is geodesic 2-bipancyclic, if every pair of vertices of it is contained in a -geodesic cycle of length for each even integer satisfying where is the distance between and In this paper, we prove that the Cartesian product of two geodesic hamiltonian graphs is a geodesic 2-bipancyclic graph. As a consequence, we show that for every -dimensional torus is a geodesic 2-bipancyclic graph
An upper bound for the crossing number of augmented cubes
A {\it good drawing} of a graph is a drawing where the edges are
non-self-intersecting and each two edges have at most one point in common,
which is either a common end vertex or a crossing. The {\it crossing number} of
a graph is the minimum number of pairwise intersections of edges in a good
drawing of in the plane. The {\it -dimensional augmented cube} ,
proposed by S.A. Choudum and V. Sunitha, is an important interconnection
network with good topological properties and applications. In this paper, we
obtain an upper bound on the crossing number of less than
.Comment: 39 page