A new approach towards a conjecture on intersecting three longest paths

Abstract

AbstractIn1966,T.Gallaiaskedwhethereveryconnectedgraphhasavertexthatappears in all longest paths. Since then this question has attracted muchattention and many work has been done in this topic. One important openquestion in this area is to ask whether any three longest paths contains acommon vertex inaconnected graph. Itwas conjectured that theanswer tothis question is positive. In this paper, we propose a new approach in viewofdistancesamonglongestpathsinaconnectedgraph,andgiveasubstantialprogresstowardstheconjecturealongtheidea. 1 Introduction In [4] Gallai asked whether every connected graph has a vertex that appears in alllongest paths. This question has attracted much attention and many work has beendone around this area of study. The answer to this question is false as stated; actuallyseveral counterexamples were given in [8, 9, 10]. A graph G is hypotraceable if Ghas no Hamiltonian path but every vertex-deleted subgraph G− v has. Note thathypotraceable graphs constitute a large class of counterexamples. Thomassen [7]showed that there exist infinitely many planar hypotraceable graphs, meaning thatthere exist infinitely many counterexamples towards the question.Yet there areclasses ofgraphs forwhich the answer toGallai’s question is positive.To see this, note that, in a tree, all longest paths must contain its center(s). Klav˘zarand Petkov˘sek [6] showed that the answer is also positive for split graphs, cacti, andsome other classes of graphs. Balister et al. [2] obtained a similar result for the classof circular arc graphs.Regarding Gallai’s question, what happens if we consider the intersection of asmaller number of longest paths? While we can easily check that every two longest