A note on extending Bondy's meta-conjecture

A graph of order n ≥ 3 is said to be pancyclic if it contains a cycle of each length from 3 to n. A chord of a cycle is an edge between two nonadjacent vertices of the cycle. A chorded cycle is a cycle containing at least one chord. We define a graph of order n ≥ 4 to be chorded pancyclic if it contains a chorded cycle of each length from 4 to n. In this article, we prove the following: If G is a graph of order n ≥ 4 with degG(x) + degG(y) ≥ n for each pair of nonadjacent vertices x, y in G, then G is chorded pancyclic, or G = Kn/2,n/2, or G is one particular small order exception. We also show this result is sharp, both in terms of the degree sum condition and in terms of the number of chords we can guarantee exist per cycle. We further extend Bondy’s meta-conjecture on pancyclic graphs to a meta-conjecture on chorded pancyclic graphs.