## On the intersections of longest cycles in a graph.(English)Zbl 0841.05056

Summary: We confirm a conjecture, due to Grötschel, regarding the intersection vertices of two longest cycles in a graph; see M. Grötschel [Graph theory and combinatorics, Proc. Conf. Hon. P. Erdös, Cambridge 1983, 171-189 (1984; Zbl 0549.05040)]. In particular, we show that if $$G$$ is a graph of circumference at least $$k+ 1$$, where $$k\in \{6, 7\}$$, and $$G$$ has two longest cycles meeting in a set $$W$$ of $$k$$ vertices, then $$W$$ is an articulation set. Grötschel had previously proved this result for $$k\in \{3, 4, 5\}$$ and shown that it fails for $$k> 7$$. As corollaries, we obtain results regarding the minimum lengths of longest cycles in certain vertex-transitive graphs. Our proofs are novel in that they make extensive use of a computer, although the programs themselves are straightforward.

### MSC:

 05C38 Paths and cycles

### Keywords:

intersection; longest cycles

Zbl 0549.05040

Mathematica
Full Text:

### References:

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