Abstract not found

In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless embeddability of G is characterized by the inequality (G) 4. In this paper we give an overview of results on (G) and of techniques to handle it.

We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3-coloring conjecture, and Pfaffian orientations of bipartite graphs.

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Let t ≥ 1 be an integer, and let G be a graph on n vertices with no minor isomorphic to Kt+1. Kostochka conjectures that there exists a constant c = c(k) independent of G such that the complement of G has a minor isomorphic to Ks, wheres=⌈1 2 (1 + 1/t)n − c⌉. We prove that Kostochka’s conjecture is equivalent to the conjecture of Duchet and Meyniel that every graph with no minor isomorphic to Kt+1 has an independent set of size at least n/t. We deduce that Kostochka’s conjecture holds for all integers t ≤ 5, and that a weaker form with s replaced by s ′ = ⌈ 1 2 (1 + 1/(2t))n − c⌉ holds for all integers t ≥ 1.