Vertex pancyclicity in quasi claw-free graphs

This paper generalizes the concept of locally connected graphs. A graph G   is triangularly connected if for every pair of edges e1,e2∈E(G)e1,e2∈E(G), G   has a sequence of 3-cycles C1,C2,…,ClC1,C2,…,Cl such that e1∈C1,e2∈Cle1∈C1,e2∈Cl and E(Ci)∩E(Ci+1)≠∅E(Ci)∩E(Ci+1)≠∅ for 1⩽i⩽l-11⩽i⩽l-1. In this paper, we show that every triangularly connected quasi claw-free graph on at least three vertices is vertex pancyclic. Therefore, the conjecture proposed by Ainouche is solved.