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Graphs and Combinatorics

, Volume 34, Issue 6, pp 1565–1580 | Cite as

Chorded Pancyclicity in k-Partite Graphs

  • Daniela FerreroEmail author
  • Linda Lesniak
Original Paper
  • 107 Downloads

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.

Keywords

Hamiltonicity Pancyclicity Bipancyclicity Chorded pancycliclity Bipartite graphs k-Partite graphs 

Mathematics Subject Classification

05C45 

Notes

Acknowledgements

We thank IPFW for its hospitality and Jay Bagga, Lowell Beineke and Marc Lipman for their comments at the beginning of this project. We thank the MAA—Tensor Program for Women and Mathematics for facilitating a research visit during this project. We also thank the anonymous referee who provided insightful comments, brought to our attention the work in [4] and suggested the study of chorded pancyclicity.

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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTexas State UniversitySan MarcosUSA
  2. 2.Department of MathematicsWestern Michigan UniversityKalamazooUSA

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