Order of the smallest counterexample to Gallai’s conjecture

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 α′(G) ≤ 5, which implies that Zamfirescu’s conjecture is true.