Properties of algebraically defined edge-colored graphs

In this paper, we introduce and investigate d -dimensional algebraically defined edge-colored (ADC) graphs, which are constructed as follows. Given a field F , and polynomials f 2 , … , f d : F 2 → F , we start with a complete bipartite graph K where both partite sets P and L are copies of F d . We color an edge between two vertices ( a , a 2 , … , a d ) ∈ P and [ x , x 2 , … , x d ] ∈ L solid blue if a i + x i = f i ( a , x ) for all 2 ≤ i ≤ d , otherwise the corresponding edge is colored dotted red. We study the existence and length of properly colored cycles and properly connected circuits, and classify ADC graphs by proper diameter. The induced subgraph generated by the set of all blue edges is an algebraically defined graph; algebraically defined graphs are of interest in the literature due to both their graph theoretic properties and their connections to incidence geometry.