#### Document Type

Article

#### Publication Date

2017

#### Department

Mathematics, Mathematics and Computer Science,Electronic and Computer Engineering

#### Abstract

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 deg_{G}(x) + deg_{G}(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.

#### Recommended Citation

Cream, Megan; Gould, Ronald J.; and Hirohata, Kazuhide, "A note on extending Bondy’s meta-conjecture" (2017). *Spelman College Faculty Publications*. 19.

http://digitalcommons.auctr.edu/scpubs/19

#### Source

Australasian Journal of Combinatorics Volume 67(3) (2017), Pages 463-469 A

## Comments

https://ajc.maths.uq.edu.au/pdf/67/ajc_v67_p463.pdf