Byeong Moon Kim

CLASSIFICATION OF TWO-REGULAR DIGRAPHS WITH MAXIMUM DIAMETER

The Klee-Quaife problem is finding the minimum order μ(d, c, v) of the (d, c, v) graph, which is a c-vertex connected v-regular graph with diameter d. Many authors contributed finding μ(d, c, v) and they also enumerated and classified the graphs in several cases. This problem is naturally extended to the case of digraphs. So we are interested in the extended Klee-Quaife problem. In this paper, we deal with an equivalent problem, finding the maximum diameter of digraphs with given order, focused on 2-regular case. We show that the maximum diameter of strongly connected 2-regular digraphs with order n is n − 3, and classify the digraphs which have diameter n− 3. All 15 nonisomorphic extremal digraphs are listed.

The exponent of Cartesian product of cycles

Abstract A digraph D is primitive if for each pair of vertices v , w of D , there is a positive integer k such that there is a directed walk of length k from v to w . The minimum of such k is the exponent of D . In this paper, we show that for a primitive graph G and a strongly connected bipartite digraph D , the exponent of the Cartesian product G × D is equal to the addition of the exponent of G and the diameter of D . Finally, we find the exponents of Cartesian products of cycles.

EXPONENTS AND DIAMETERS OF STRONG PRODUCTS OF DIGRAPHS

The exponent of the strong product of a digraph of order m and a digraph of order n is shown to be bounded above by m+ n 2, with equality for Zm ⊠ Zn. The exponent and diameter of the strong product of a graph and a digraph are also investigated.