Mathematics 5707 Homework 1

number of ordered triples (a, b, c) such that ab ∈ E(G) and bc / ∈ E(G). Observe that each vertex v has deg v = 0, 1, 2, 3, 4, or 5. If deg v is 0 or 5, then v is the second vertex in 0 such triples. If deg v is 1 or 4, then v is the second vertex in 4 such triples. If deg v is 2 or 3, then v is the second vertex in 6 such triples. Thus, there are at most 36 such ordered triples. Note that each unordered triple of vertices that does not form a triangle-or-antitriangle corresponds to two such ordered triples, so at most 18 unordered triples of vertices do not form triangle-or-antitriangles, so there are at least two triangle-or-antitriangles in G.