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\((r)\)-pancyclic, \((r)\)-bipancyclic and oddly \((r)\)-bipancyclic graphs. (English) Zbl 1377.05092

Summary: A graph with \(v\) vertices is \((r)\)-pancyclic if it contains precisely \(r\) cycles of every length from 3 to \(v\). A bipartite graph with even number of vertices \(v\) is said to be \((r)\)-bipancyclic if it contains precisely \(r\) cycles of each even length from 4 to \(v\). A bipartite graph with odd number of vertices \(v\) and minimum degree at least 2 is said to be oddly \((r)\)-bipancyclic if it contains precisely \(r\) cycles of each even length from 4 to \(v-1\).
In this paper, using a computer search, we classify all \((r)\)-pancyclic and \((r)\)-bipancyclic graphs, \(r\geq 2\), with \(v\) vertices and at most \(v+5\) edges. We also classify all oddly \((r)\)-bipancyclic graphs, \(r\geq 1\), with \(v\) vertices and at most \(v+4\) edges.

MSC:

05C38 Paths and cycles
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Full Text: arXiv