CHARACTERIZATION OF A CLASS OF TRIANGLE-FREE GRAPHS WITH A CERTAIN ADJACENCY PROPERTY

Let m and n be nonnegative integers. Denote by P(m,n) the set of all triangle-free graphs G such that for any independent m-subset M and any n-subset N of V(G) with M intersect N = empty-set, there exists a unique vertex of G that is adjacent to each vertex in M and nonadjacent to any vertex in N. We prove that if m greater-than-or-equal-to 2 and n greater-than-or-equal-to 1, then P(m,n) = empty-set whenever m less-than-or-equal-to n, and P(m,n) = {K(m,n + 1)} whenever m > n. We also have P(1,1) = {C5} and P(1,n) = empty-set for n greater-than-or-equal-to 2. In the degenerate cases, the class P(0,n) is completely determined, whereas the class P(m,0), which is most interesting, being rich in graphs, is partially determined.