Functional graphs

Abstract

A graph G is said to be a functional graph if there exist two mappings f and g from V ( G ) into a set F such that xy is an edge in G whenever f ( x ) = g ( y ) or g ( x ) = f ( y ) . Chvátal and Ebenegger proved that recognizing functional graphs is an NP-complete problem. Using the compactness theorem, we prove that if G is an infinite graph such that any finite subgraph of G is a functional graph, then G is a functional graph. We give an elementary proof of this fact in the infinite countable case. In the finite case, we prove that for n large enough, any graph of girth n containing at most 3 n − 7 vertices is a functional graph. It will be shown by an example that this bound is the best possible. To cite this article: A. El Sahili, C. R. Acad. Sci. Paris, Ser. I 341 (2005).