Choudum and Sunitha [Networks 40 (2002) 71–84] proposed the class of augmented cubes as a variation of hypercubes and showed that augmented cubes possess several embedding properties that the hypercubes and other variations do not possess. Recently, Hsu et al. [Information Processing Letters 101 (2007) 227–232] showed that the ndimensional augmented cube AQn, n 2, is weakly geodesic-pancyclic, ...

Given a k-arc-strong tournament T , we estimate the minimum number of arcs possible in a k-arc-strong spanning subdigraph of T . We give a construction which shows that for each k ≥ 2 there are tournaments T on n vertices such that every k-arc-strong spanning subdigraph of T contains at least nk+ k(k−1) 2 arcs. In fact, the tournaments in our construction have the property that every spanning s...

An in-tournament is an oriented graph such that the in-neighborhood of every vertex induces a tournament. Therefore, in-tournaments are a generalization of local tournaments where, for every vertex, the set of inneighbors as well as the set of out-neighbors induce a tournament. While local tournaments have been intensively studied very little is known about in-tournaments. It is the purpose of ...

Bondy proved that an n-vertex simple Hamiltonian graph with at least n2/4 edges has cycles of every length unless it is isomorphic to Kn/2,n/2. This paper considers finding circuits of every size in GF (q)-representable matroids with large numbers of elements. A consequence of the main result is that a rank-r simple binary matroid with at least 2r−1 elements has circuits of all sizes or is isom...