Abstract
In 1966, Gallai asked whether all longest paths in a connected graph share a common vertex. Counterexamples indicate that this is not true in general. However, Gallai's question is positive for certain well-known classes of connected graphs, such as split graphs, interval graphs, circular arc graphs, outerplanar graphs, and series-parallel graphs. A graph is $2K_2$-free if it does not contain two independent edges as an induced subgraph. In this short note, we show that, in nonempty $2K_2$-free graphs, every vertex of maximum degree is common to all longest paths. Our result implies that all longest paths in a nonempty $2K_2$-free graph have a nonempty intersection. In particular, it strengthens the result on split graphs, as split graphs are $2K_2$-free.
Highlights
The question was first asked by Zamfirescu [21]. It is mentioned in [4, 11, 24, 26]. Progress in this direction was made by de Rezende et al [8], who proved that if all nontrivial blocks of a connected graph are Hamiltonian, any three longest paths in the graph share a vertex
We investigate the intersection of all longest paths in connected 2K2-free graphs
The class of 2K2-free graphs is well studied, for instance, see [3, 6, 9, 14, 15, 16, 18]. It is a superclass of the class of split graphs, where vertices can be partitioned into a clique and an independent set
Summary
All graphs considered in this paper are finite and simple. A path P in a graph G is a longest path in G if there is no path in G strictly longer than P. It is well-known that in a connected graph, any two longest paths have a vertex in common [17]. This motivated Gallai in 1966 to ask whether all longest paths in a connected graph have a vertex in common. A connected graph share a common vertex.
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