On the characterization of some algebraically defined bipartite graphs of girth eight

Abstract

For any field F and polynomials f2,f3∈F[x,y], let ΓF(f2,f3) denote the bipartite graph with vertex partition P∪L, where P and L are two copies of F3, and (p1,p2,p3)∈P is adjacent to [l1,l2,l3]∈L if and only if p2+l2=f2(p1,l1) and p3+l3=f3(p1,l1). The graph Γ3(F)=ΓF(xy,xy2) is known to be of girth eight. When F=Fq is a finite field of odd characteristic or F=F∞ is an algebraically closed field of characteristic zero, the graph Γ3(F) is conjectured to be the unique one with girth at least eight among those ΓF(f2,f3) up to isomorphism. This conjecture has been confirmed for the case that both f2,f3 are monomials over Fq, and for the case that at least one of f2,f3 is a monomial over F∞. If one of f2,f3∈Fq[x,y] is a monomial, it has also been proved the existence of a positive integer M such that G=ΓFqM(f2,f3) is isomorphic to Γ3(FqM) provided G has girth at least eight. In this paper, these results are shown to be valid when the restriction on the polynomials f2,f3 is relaxed further to that one of them is the product of two univariate polynomials. Furthermore, all of such polynomials f2,f3 are characterized completely.