On the girth of two-dimensional real algebraically defined graphs

Abstract

For a polynomial f∈R[X,Y], we define a bipartite graph ΓR(f) where each partite set is a copy of R2. Furthermore, (a1,a2) in the first partite set is adjacent to [x1,x2] in the second if and only if a2+x2=f(a1,x1). The main result of this paper is that every graph ΓR(f) has girth 4 or 6, and moreover we classify infinite families of such graphs by girth.