International Journal of Mathematics and Mathematical Sciences

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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 pk, if it exists, such that whenever G is a rank k graph of order p≤pk 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 pk exists. It is further shown that for k=0, 1, and 2, pk=5, 6, and 7, respectively.