Elsevier

Discrete Mathematics

Volume 341, Issue 8, August 2018, Pages 2131-2141
Discrete Mathematics

Cycles with a chord in dense graphs

https://doi.org/10.1016/j.disc.2018.04.016Get rights and content
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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 n4. Then a graph G of order n is chorded pancyclic if G contains a chorded k-cycle for every integer k with 4kn. Cream, Gould and Hirohata (Australas. J. Combin. 67 (2017), 463–469) proved that a graph of order n satisfying degGu+degGvn for every pair of nonadjacent vertices u,  v in G is chorded pancyclic unless G is either Kn2,n2 or K3K2, 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 K3K2, Then observing that Kn2,n2 and K3K2 are exceptions only for k=4, we further relax the density condition for sufficiently large k.

Keywords

Pancyclic
Weakly pancyclic
Chorded pancyclic
Chorded cycle

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