Graphs which have pancyclic complements

H. Joseph Straight

International Journal of Mathematics and Mathematical Sciences, 1978, vol. 1, 1-9

Abstract:

Let p and q denote the number of vertices and edges of a graph G , respectively. Let Δ ( G ) denote the maximum degree of G , and G ¯ the complement of G . A graph G of order p is said to be pancyclic if G contains a cycle of each length n , 3 ≤ n ≤ p . For a nonnegative integer k , a connected graph G is said to be of rank k if q = p − 1 + k . (For k equal to 0 and 1 these graphs are called trees and unicyclic graphs, respectively.)

In 1975, I posed the following problem: Given k , find the smallest positive integer p k , if it exists, such that whenever G is a rank k graph of order p ≤ p k and Δ ( G ) < p − 2 then G ¯ is pancyclic. In this paper it is shown that a result by Schmeichel and Hakiml (2) guarantees that p k exists. It is further shown that for k = 0 , 1 , and 2 , p k = 5 , 6 , and 7 , respectively.

Date: 1978
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:380981

DOI: 10.1155/S0161171278000216

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