Czechoslovak Mathematical Journal, first online, pp. 1-29


Order of the smallest counterexample to Gallai's conjecture

Fuyuan Chen

Received August 7, 2016.   First published February 7, 2018.

Abstract:  In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Zamfirescu conjectured that the smallest counterexample to Gallai's conjecture is a graph on 12 vertices. We prove that Gallai's conjecture is true for every connected graph $G$ with $\alpha'(G)\leq5$, which implies that Zamfirescu's conjecture is true.
Keywords:  longest path; matching number
Classification MSC:  05C38, 05C70, 05C75
DOI:  10.21136/CMJ.2018.0422-16

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Affiliations:   Fuyuan Chen, Anhui University of Finance and Economics, No. 962 Caoshan Road, Bengbu City, P.R. China, e-mail: chenfuyuan19871010@163.com


 
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