Lattices of properties of countable graphs and the Hedetniemi Conjecture

Abstract

Lattices of hereditary properties of nite graphs have been extensively studied. We investigate the lattice L of induced-hereditary properties of countable graphs. Of interest to us will be some of the members of L. Much of our focus will be on hom-properties. We analyze their behaviour and consider their link to solving the long standing Hedetniemi Conjecture. We then discuss universal graphs and construct a universal graph for hom-properties. We then use these universal graphs to prove a theorem by Szekeres and Wilf. Lastly we off er a new proof of a theorem by Du ffus, Sands and Woodrow.