Abstract
This paper generalizes the concept of locally connected graphs. A graph G is triangularly connected if for every pair of edges e 1 , e 2 ∈ E ( G ) , G has a sequence of 3-cycles C 1 , C 2 , … , C l such that e 1 ∈ C 1 , e 2 ∈ C l and E ( C i ) ∩ E ( C i + 1 ) ≠ ∅ for 1 ⩽ 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.
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