Minimum order of r-regular bipartite graphs of pair length k

The concepts of k-pairable graphs and the pair length of a graph were introduced by Chen [Discrete Math. 287 (2004), 11-15] to generalize an elegant result of Graham et al. [Amer. Math. Monthly 101 (1994), 664- 667] from hypercubes and graphs with antipodal isomorphisms to a much larger class of graphs. A graph G is k-pairable if there is a positive integer k such that the automorphism group of G contains an involution φ with the property that the distance between x and φ(x) is at least k for any vertex x of G. The pair length of a graph G, denoted by p(G), is the maximum positive integer k such that G is k-pairable; and p(G) = 0 if G is not k-pairable for any positive integer k. The aim of this paper is to answer an open question posted in our previous paper [Discrete Math. 310 (2010), 3334-3350]; that is, the question of determining the minimum order of a graph in the set of r-regular bipartite graphs of pair length k. We solve the problem for all positive integers k and r except for the case when both k ≥ 5 and r ≥ 3 are odd. For the case that is still open, we provide bounds on the minimum order concerned. Also we post a conjecture on the minimum order of a cubic bipartite graph of pair length k for any odd number k > 1.