Proof of a conjecture on monomial graphs

Let $e$ be a positive integer, $p$ be an odd prime, $q=p^{e}$, and $\Bbb F_q$ be the finite field of $q$ elements. Let $f,g \in \Bbb F_q [X,Y]$. The graph $G=G_q(f,g)$ is a bipartite graph with vertex partitions $P=\Bbb F_q^3$ and $L=\Bbb F_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\in P$ is adjacent to a vertex $[l] = [l_1,l_2,l_3]\in L$ if and only if $p_2 + l_2 = f(p_1,l_1)$ and $p_3 + l_3 = g(p_1,l_1)$. Motivated by some questions in finite geometry and extremal graph theory, Dmytrenko, Lazebnik and Williford 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 $G_q(XY,XY^2)$. They proved several instances of the conjecture by reducing it to the property of polynomials $A_k= X^k[(X+1)^k - X^k]$ and $B_k= [(X+1)^{2k} - 1] X^{q-1-k} - 2X^{q-1}$ being permutation polynomials of $\Bbb F_q$. In this paper we prove the conjecture by obtaining new results on the polynomials $A_k$ and $B_k$, which are also of interest on their own.