| Article ID | Journal | Published Year | Pages | File Type |
|---|---|---|---|---|
| 4656922 | Journal of Combinatorial Theory, Series B | 2012 | 19 Pages |
The bull is the graph consisting of a triangle and two disjoint pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is the first paper in a series of three. The goal of the series is to explicitly describe the structure of all bull-free graphs. In this paper we study the structure of bull-free graphs that contain as induced subgraphs three-edge-paths P and Q, and vertices c∉V(P) and a∉V(Q), such that c is adjacent to every vertex of V(P) and a has no neighbor in V(Q). One of the theorems in this paper, namely 1.2, is used in Chudnovsky and Safra (2008) [9], in order to prove that every bull-free graph on n vertices contains either a clique or a stable set of size , thus settling the Erdös–Hajnal conjecture (Erdös and Hajnal, 1989) [17] for the bull.
