Minimum cycle bases of direct products of complete graphs

Minimum cycle bases for direct products of K2 with complete graphs

We construct a minimum cycle basis for the direct product K2 × Kp of two complete graphs where p ≥ 2. For p > 3, these bases consists only of squares. This completes the work of R. Hammack, who treated the case Kp ×Kq for p, q ≥ 3 [Inform. Process. Lett. 102 (2007), 214–218.]

Minimum cycle bases of direct products of graphs with cycles

We construct a minimum cycle basis for the direct product G × Cq where G is a connected non-bipartite graph, Cq is an odd cycle and G×Cq is triangle-free. These bases are expressed in terms of the cycle structure of the symmetric digraph on G.

Minimum cycle bases of direct products of bipartite graphs

Minimum cycle bases of lexicographic products

Minimum cycle bases of product graphs can in most situations be constructed from minimum cycle bases of the factors together with a suitable collection of triangles and/or quadrangles determined by the product operation. Here we give an explicit construction for the lexicographic product G ◦ H that generalizes results by Berger and Jaradat to the case that H is not connected.

Minimum cycle bases of product graphs

A construction for a minimal cycle basis for the Cartesian and the strong product of two graphs from the minimal length cycle bases of the factors is presented. Furthermore, we derive asymptotic expressions for the average length of the cycles in the minimal cycle bases of the powers (iterated products) of graphs. In the limit only triangles and squares play a role.