A number theoretic problem on super line graphs

In Bagga et al. (1995) a generalization of the line graph concept was introduced. Given a graph GG with at least rr edges, the super line graph of index  rr, Lr(G)Lr(G), has as its vertices the sets of rr edges of GG, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number  lc(G) of a graph GG is the least index rr for which Lr(G)Lr(G) is complete. In this paper we investigate the line completion number of Km,nKm,n. This turns out to be an interesting optimization problem in number theory, with results depending on the parities of mm and nn. If m≤nm≤n and mm is a fixed even number, then lc(Km,n) has been found for all even values of nn and for all but finitely many odd values. However, when mm is odd, the exact value of lc(Km,n) has been found in relatively few cases, and the main results concern lower bounds for the parameter. Thus, the general problem is still open, with about half of the cases unsettled.