Proof of a conjecture on monomial graphs

Let e be a positive integer, p be an odd prime, q = p, and Fq be the finite field of q elements. Let f, g ∈ Fq[X,Y ]. The graph G = Gq(f, g) is a bipartite graph with vertex partitions P = F3q and L = F 3 q, and edges defined as follows: a vertex (p) = (p1, p2, p3) ∈ P is adjacent to a vertex [l] = [l1, l2, l3] ∈ L if and only if p2 + l2 = f(p1, l1) and p3 + l3 = g(p1, l1). Motivated by some questions in finite geometry and extremal graph theory, Dmytrenko, Lazebnik andWilliford conjectured in 2007 that if f and g are both monomials and G has no cycle of length less than eight, then G is isomorphic to the graph Gq(XY,XY ). They proved several instances of the conjecture by reducing it to the property of polynomials Ak = X [(X + 1) − X] and Bk = [(X + 1) 2k − 1]X − 2X being permutation polynomials of Fq. In this paper we prove the conjecture by obtaining new results on the polynomials Ak and Bk, which are also of interest on their own.