Proof of a conjecture on monomial graphs

Abstract

Abstract. Let e be a positive integer, p be an odd prime, q = p e , 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 = F 3 q and L = F 3 q , and edges defined as follows: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 Gq(XY, XY 2 ). They proved several instances of the conjecture by reducing it to the property of polynomialsIn 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.

Classification by girth of three-dimensional algebraically defined monomial graphs over the real numbers

“…The primary goal of study in [5] was to ascertain that when R is a finitehttps://www.overleaf.com/project/5db074df0a3a250001 field F q of odd order q, the only (up to isomorphism) girth eight graph Γ R (f, g), where f and g are monomials in R[X, Y ], is Γ R (XY, XY 2 ). A similar assertion was proven in [11] and [12]: whenever R is an algebraically closed field of characteristic zero, the only (up to isomorphism) graph Γ R (X k Y m , g) of girth at least eight, where k, m ∈ N and g ∈ R[X, Y ], is Γ R (XY, XY 2 ).…”

“…Of particular interest is the question whether all such graphs are isomorphic to Γ R (XY, XY 2 ); this stands in contrast to the situation R = F q with q odd (as in [5]) and R = C (as in [11,12]), where it is known that all monomial graphs of girth eight are isomorphic to Γ R (XY, XY 2 ). We do not for instance know whether Γ R (XY, XY 2 ) is isomorphic to Γ R (XY, XY 4 ).…”

“…They conjectured that all such monomial graphs of girth at least eight are isomorphic to Γ Fq (XY, XY 2 ). This work was expanded upon by Kronenthal [10], and the conjecture was ultimately proven by Hou, Lappano, and Lazebnik [5]. In addition, Kronenthal and Lazebnik [11] and Kronenthal, Lazebnik, and Williford [12] studied families of polynomial graphs over algebraically closed fields of characteristic zero and applied some of their techniques to graphs over finite fields; these results were recently extended by Xu, Cheng, and Tang [15].…”

Bipartite algebraic graphs without quadrilaterals

“…Also, a conjecture in a similar spirit about algebraic graphs of girth eight was made by Dmytrenko, Lazebnik and Williford [DLW07]. It was recently resolved by Hou, Lappano and Lazebnik [HLL17]. 4 Henceforth, a statement is true for a generic point u ∈ P s means that there exists a nonempty Zariski-open set U ⊂ P s such that the statement is true for every u ∈ U .…”

On the DLW conjectures

“…Although the polynomials A k and B k are both related to the graph G q (XY, X k Y 2k ), it is not clear how they are related to each other. Therefore, it is natural to consider the polynomials A k and B k separately, giving rise to the following two stronger versions of Conjecture 2; see [4,5,7].…”