Chorded Pancyclicity in k-Partite Graphs

Abstract

We prove that for any integers $$p\ge k\ge 3$$ and any k-tuple of positive integers $$(n_1,\ldots ,n_k)$$ such that $$p=\sum _{i=1}^k{n_i}$$ and $$n_1\ge n_2\ge \cdots \ge n_k$$ , the condition $$n_1\le {p\over 2}$$ is necessary and sufficient for every subgraph of the complete k-partite graph $$K(n_1,\ldots ,n_k)$$ with at least $$\begin{aligned} {{4 -2p+2n_1+\sum _{i=1}^{k} n_i(p-n_i)}\over 2} \end{aligned}$$ edges to be chorded pancyclic. Removing all but one edge incident with any vertex of minimum degree in $$K(n_1,\ldots ,n_k)$$ shows that this result is best possible. Our result implies that for any integers, $$k\ge 3$$ and $$n\ge 1$$ , a balanced k-partite graph of order kn with has at least $${{(k^2-k)n^2-2n(k-1)+4}\over 2}$$ edges is chorded pancyclic. In the case $$k=3$$ , this result strengthens a previous one by Adamus, who in 2009 showed that a balanced tripartite graph of order 3n, $$n \ge 2$$ , with at least $$3n^2 - 2n + 2$$ edges is pancyclic.