Geodesic
Chorded $k$-pancyclic and weakly $k$-pancyclic graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 337-350

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As natural relaxations of pancyclic graphs, we say a graph G is k-pancyclic if G contains cycles of each length from k to |V(G)| and G is weakly pancyclic if it contains cycles of all lengths from the girth to the circumference of G, while G is weakly k-pancyclic if it contains cycles of all lengths from k to the circumference of G. A cycle C is chorded if there is an edge between two vertices of the cycle that is not an edge of the cycle. Combining these ideas, a graph is chorded pancyclic if it contains chorded cycles of each length from 4 to the circumference of the graph, while G is chorded k-pancyclic if there is a chorded cycle of each length from k to |V(G)|. Further, G is chorded weakly k-pancyclic if there is a chorded cycle of each length from k to the circumference of the graph. We consider conditions for graphs to be chorded weakly k-pancyclic and chorded k-pancyclic.
Keywords: cycle, chord, pancyclic, weakly pancyclic 
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Cream, Megan; Gould, Ronald J. Chorded $k$-pancyclic and weakly $k$-pancyclic graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 337-350. http://geodesic.mathdoc.fr/item/DMGT_2024_44_1_a16/