Cycles with a chord in dense graphs

Abstract A cycle of order k is called a k -cycle. A non-induced cycle is called a chorded cycle. Let n be an integer with n ≥ 4 . Then a graph G of order n is chorded pancyclic if G contains a chorded k -cycle for every integer k with 4 ≤ k ≤ n . Cream, Gould and Hirohata (Australas. J. Combin. 67 (2017), 463–469) proved that a graph of order n satisfying deg G u + deg G v ≥ n for every pair of nonadjacent vertices u , v in G is chorded pancyclic unless G is either K n 2 , n 2 or K 3 □ K 2 , the Cartesian product of K 3 and K 2 . They also conjectured that if G is Hamiltonian, we can replace the degree sum condition with the weaker density condition | E ( G ) | ≥ 1 4 n 2 and still guarantee the same conclusion. In this paper, we prove this conjecture by showing that if a graph G of order n with | E ( G ) | ≥ 1 4 n 2 contains a k -cycle, then G contains a chorded k -cycle, unless k = 4 and G is either K n 2 , n 2 or K 3 □ K 2 , Then observing that K n 2 , n 2 and K 3 □ K 2 are exceptions only for k = 4 , we further relax the density condition for sufficiently large k .