Let G be a graph on n vertices, with maximal degree d, and not containing K1,k as an induced subgraph. We prove: 1. λ(G) ≤ (2-1/2k-2+o(1))d 2. η(I(G))≥n(k-1)/d(2k-3)+k-1. Here λ(G) is the maximal eigenvalue of the Laplacian of G, I(G) is the independence complex of G, and η(C) denotes the topological connectivity of a complex C plus 2. These results provide improved bounds for the existence of independent transversals in K1,k-free graphs.
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- homological connectivity
- independent transversals