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Discussiones Mathematicae Graph Theory 29(1) (2009) 5-14
DOI: https://doi.org/10.7151/dmgt.1429

STRONGLY PANCYCLIC AND DUAL-PANCYCLIC GRAPHS

Terry A. McKee

Department of Mathematics & Statistics
Wright State University
Dayton, Ohio 45435, USA

Abstract

Say that a cycle C almost contains a cycle C⁻ if every edge except one of C⁻ is an edge of C. Call a graph G strongly pancyclic if every nontriangular cycle C almost contains another cycle C⁻ and every nonspanning cycle C is almost contained in another cycle C⁺. This is equivalent to requiring, in addition, that the sizes of C⁻ and C⁺ differ by one from the size of C. Strongly pancyclic graphs are pancyclic and chordal, and their cycles enjoy certain interpolation and extrapolation properties with respect to almost containment. Much of this carries over from graphic to cographic matroids; the resulting `dual-pancyclic' graphs are shown to be exactly the 3-regular dual-chordal graphs.

Keywords: pancyclic graph, cycle extendable, chordal graph, pancyclic matroid, dual-chordal graph.

2000 Mathematics Subject Classification: 05C38 (05C62, 05C45).

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Received 23 May 2007
Revised 26 November 2008
Accepted 26 November 2008