International Mathematical Forum, Vol. 7, 2012, no. 54, 2675 - 2680 The Fair Sharing Graph and its Helly Property Jonald P. Fenecios Department of Mathematics Ateneo de Davao University E. Jacinto St., Davao City, Philippines 8000 jpfenecios@addu.edu.ph Abraham P. Racca Department of Mathematics and Physics Adventist University of the Philippines Puting Kahoy, Silang Cavite, The Philippines abraham.racca@yahoo.com Abstract We define a new class of graphs called fair sharing graphs and prove that every three longest paths in a fair sharing graph share a common vertex. This verifies that the well-known conjecture that every three longest paths in a connected graph share a common vertex is true in this class of graphs. Mathematics Subject Classification: 05C38 Keywords: Fair sharing graph, Helly Property 1 Introduction In this study, we define a new class of graphs called fair sharing graphs. This is in connection with the question posed by Gallai in 1966 who asked whether every connected graph has a vertex that appears in all of its longest paths. Subsequent developments of the theory have shown that Gallai’s conjecture is not true in general. H. Walter found the first counterexample of the conjec- ture and later Zamfirescu found a graph with 12 vertices in which there is no common vertex. However, if one asks whether every pair of longest paths in a connected graph share a common vertex the answer is surprisingly in the