Fractional hypergraph isomorphism and fractional invariants
share
Fractional graph isomorphism is the linear relaxation of an integer programming formulation of graph isomorphism. It preserves some invariants of graphs, like degree sequences and equitable partitions, but it does not preserve others like connectivity, clique and independence numbers, chromatic number, vertex and edge cover numbers, matching number, domination and total domination numbers. In this work, we extend the concept of fractional graph isomorphism to hypergraphs, and give an alternative characterization, analogous to one of those that are known for graphs. With this new concept we prove that the fractional packing, covering, matching and transversal numbers on hypergraphs are invariant under fractional hypergraph isomorphism. As a consequence, fractional matching, vertex and edge cover, independence, domination and total domination numbers are invariant under fractional graph isomorphism. This is not the case of fractional chromatic, clique, and clique cover numbers. In this way, most of the classical fractional parameters are classified with respect to their invariance under fractional graph isomorphism.
READ FULL TEXT