An even-hole-free graph is a graph with no induced cycle of even length. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. Reed conjectured in  that every nonnull even-hole-free graph has a bisimplicial vertex. The authors published a paper  in which they claimed a proof, but there is a serious mistake in that paper, recently brought to our attention by Rong Wu. The error in  is in the last line of the proof of theorem 3.1 of that paper: we say “it follows that [Formula presented], and so v is bisimplicial in G”; and this is not correct, since cliques of [Formula presented] may not be cliques of G. Unfortunately, the flawed theorem 3.1 is fundamental to much of the remainder of the paper, and we have not been able to fix the error (although we still believe 3.1 to be true). Thus, this paper does not prove Reed's conjecture after all. Two of us (Chudnovsky and Seymour) claim to have a proof of Reed's conjecture using a different method .
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics