Algebraic Matching Theory

The number of vertices missed by a maximum matching in a graph G is the multiplicity of zero as a root of the matchings polynomial μ(G,x) of G, and hence many results in matching theory can be expressed in terms of this multiplicity. Thus, if mult(θ,G) denotes the multiplicity of θ as a zero of μ(G,x), then Gallai’s lemma is equivalent to the assertion that if mult(θ,G\u) < mult(θ,G) for each vertex u of G, then mult(θ, G) = 1. This paper extends a number of results in matching theory to results concerning mult(θ,G), where θ is not necessarily zero. If P is a path in G then G \ P denotes the graph got by deleting the vertices of P from G. We prove that mult(θ,G \P ) ≥ mult(θ,G)− 1, and we say P is θ-essential when equality holds. We show that if, all paths in G are θ-essential, then mult(θ,G) = 1. We define G to be θ-critical if all vertices in G are θ-essential and mult(θ,G) = 1. We prove that if mult(θ, G) = k then there is an induced subgraph H with exactly k θ-critical components, and the vertices in G\H are covered by k disjoint paths. AMS Classification Numbers: 05C70, 05E99 1 Support from grant OGP0009439 of the National Sciences and Engineering Council of Canada is gratefully acknowledged.