On the helly property working as a compactness criterion on graphs

Abstract

The study of combinatorial topology and of the most important methods in algebraic topology (simplicial complexes, discretization) leads to the idea that it may be useful to translate some of the most classical problems in topology into a discrete context. Following this principle, several authors have already tried to study fixed point and retraction problems inside the theory of partially ordered sets. We try here to make a special study about the extension of homomorphisms and the fixed point problems on graphs. We introduce here, using the Helly property, a kind of compactness tool working on graphs, and we prove a generalization of Sperner's lemma which is used in the proof of the Brouwer fixed-point theorem by Kuratowski.