Eigenvalues and Perfect Matchings

AMS classification: 05C50, 05C70, 05E30.graph;perfect matching;Laplacian matrix;eigenvalues.

A Note on Near-factor-critical Graphs

A near-factor of a finite simple graph $G$ is a matching that saturates all vertices except one. A graph $G$ is said to be near-factor-critical if the deletion of any vertex from $G$ results in a subgraph that has a near-factor. We prove that a connected graph $G$ is near-factor-critical if and only if it has a perfect matching. We also characterize disconnected near-factor-critical graphs.Comment: 4 page

Star-factors of tournaments

Let S_m denote the m-vertex simple digraph formed by m-1 edges with a common tail. Let f(m) denote the minimum n such that every n-vertex tournament has a spanning subgraph consisting of n/m disjoint copies of S_m. We prove that m lg m - m lg lg m <= f(m) <= 4m^2 - 6m for sufficiently large m.Comment: 5 pages, 1 figur