On the Baer–Lovász–Tutte construction of groups from graphs: Isomorphism types and homomorphism notions

Abstract

Let p be an odd prime. From a simple undirected graph G, through the classical procedures of Baer (1938), Tutte (1947) and Lovász (1989), there is a p-group PG of class 2 and exponent p that is naturally associated with G. Our first result is to show that this construction of groups from graphs respects isomorphism types. That is, given two graphs G and H, G and H are isomorphic as graphs if and only if PG and PH are isomorphic as groups. Our second contribution is a new homomorphism notion for graphs. Based on this notion, a category of graphs can be defined, and the Baer–Lovász–Tutte construction naturally leads to a functor from this category of graphs to the category of groups.