Intersecting longest paths

Abstract In 1966, Gallai asked whether every connected graph has a vertex that is common to all longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs and 2-trees. Another related question was raised by Zamfirescu in the 1980s: Do any three longest paths in a connected graph have a vertex in common? The answer to this question is unknown. We prove that for connected graphs in which all nontrivial blocks are Hamiltonian the answer is affirmative. Finally, we state a conjecture and explain how it relates to the three longest paths question.

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