Cycles with a chord in dense graphs

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 degGu+degGv≥n for every pair of nonadjacent vertices u,  v in G is chorded pancyclic unless G is either Kn2,n2 or K3□K2, the Cartesian product of K3 and K2. They also conjectured that if G is Hamiltonian, we can replace the degree sum condition with the weaker density condition |E(G)|≥14n2 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)|≥14n2 contains a k-cycle, then G contains a chorded k-cycle, unless k=4 and G is either Kn2,n2 or K3□K2, Then observing that Kn2,n2 and K3□K2 are exceptions only for k=4, we further relax the density condition for sufficiently large k.