TY - JOUR

T1 - Corrigendum to “Bisimplicial vertices in even-hole-free graphs” (Bisimplicial vertices in even-hole-free graphs (2008) 98(6) (1119–1164), (S0095895608000087), (10.1016/j.jctb.2007.12.006))

AU - Addario-Berry, Louigi

AU - Chudnovsky, Maria

AU - Havet, Frédéric

AU - Reed, Bruce

AU - Seymour, Paul

N1 - Funding Information:
This material is based upon work supported in part by the U. S. Army Research Office under grant number W911NF-16-1-0404, and by NSF grant DMS-1763817.Partially supported by NSF grant DMS-1800053 and AFOSR grant A9550-19-1-0187.

PY - 2020/5

Y1 - 2020/5

N2 - 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 [3] that every nonnull even-hole-free graph has a bisimplicial vertex. The authors published a paper [1] 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 [1] 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 [2].

AB - 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 [3] that every nonnull even-hole-free graph has a bisimplicial vertex. The authors published a paper [1] 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 [1] 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 [2].

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U2 - 10.1016/j.jctb.2020.02.001

DO - 10.1016/j.jctb.2020.02.001

M3 - Comment/debate

AN - SCOPUS:85079200326

VL - 142

SP - 374

EP - 375

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

ER -