Corrigendum to “Bisimplicial vertices in even-hole-free graphs”
Corrigendum to “Bisimplicial vertices in even-hole-free graphs”
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对“偶无孔图中的双单纯顶点”的勘误
DOI：
10.1016/j.jctb.2020.02.001
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Seymour, Paul
中科院分区：
文献类型：
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作者：
Addario-Berry, Louigi;Chudnovsky, Maria;Havet, Frédéric;Reed, Bruce;Seymour, Paul
偶数无孔图是不存在偶数长度的诱导环的图。如果图的顶点的邻居集合是两个派系的并集，则该图的顶点是双单纯形的。 Reed 在[3]中猜想每个非零偶无洞图都有一个双单纯顶点。作者发表了一篇论文[1]，其中声称有证明，但该论文中有一个严重错误，最近由 Rong Wu 提请我们注意。 [1]中的错误在于该论文定理3.1证明的最后一行：我们说“它遵循NG（v）= NG（v），因此v在G中是双单的”；这是不正确的，因为 G 的派系 / 可能不是 G 的派系。不幸的是，有缺陷的定理 3.1 是本文其余大部分内容的基础，并且我们无法修复该错误（尽管我们仍然相信 3.1 是正确的）。因此，这篇论文终究没有证明里德的猜想。
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 NG (v)= NG (v), and so v is bisimplicial in G”; and this is not correct, since cliques of G/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.