Comment on Unsolved Problem Number Four

Kotzig's Conjecture

Roland Häggkvist

It is certainly true that the references supplied by Yair Caro claim to settle the conjecture in total, but a closer scrutiny of the Xing and Hu paper reveals an unfortunate gap in the argument. They show in the paper, and as far as I know correctly and in my words, that if k>11 then any graph with a (2k-8)-cycle contains some pair of vertices joined by at least two paths of length (exactly) k. In other words, no counterexample to the Kotzig conjecture has a cycle of length exactly 2k-8. They then contrast this with a lemma of Kotzig, a lemma incidentally cited without proof or reference to any printed proof, which, again as I can see correctly, claims that: Any P(k)-graph contains a 2n-cycle for a n with 3\leq n\leq k-4.

From this they draw the incorrect conclusion that any P(k) graph has a (2k-8)-cycle, a contradiction that would prove the Kotzig conjecture for k>11. The conclusion is incorrect because Kotzig only in essence states that there exists some 2n cycle in the interval, not that there exists such a cycle for all n. My reading of this may of course be incorrect, but if it is then it still remains to fill in Kotzig's part of the proof.
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