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Extending vertex and edge pancyclic graphs. (English) Zbl 1402.05120
Summary: A graph $$G$$ of order $$n\geq 3$$ is pancyclic if $$G$$ contains a cycle of each possible length from 3 to $$n$$, and vertex pancyclic (edge pancyclic) if every vertex (edge) is contained on a cycle of each possible length from 3 to $$n$$. A chord of a cycle is an edge between two nonadjacent vertices of the cycle, and chorded cycle is a cycle containing at least one chord. We define a graph $$G$$ of order $$n\geq 4$$ to be chorded pancyclic if $$G$$ contains a chorded cycle of each possible length from 4 to $$n$$. In this article, we consider extensions of the property of being chorded pancyclic to chorded vertex pancyclic and chorded edge pancyclic.
##### MSC:
 05C38 Paths and cycles 05C12 Distance in graphs
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##### References:
 [1] Bondy, J.A.: Pancyclic graphs. Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing, pp. 167-172. Louisiana State Univ., Baton Rouge (1971) · Zbl 0291.05109 [2] Bondy, JA, Pancyclic graphs I, J. Combin. Theory Ser. B, 11, 80-84, (1971) · Zbl 0183.52301 [3] Chen, G.; Gould, RJ; Gu, X.; Saito, A., Cycles with a chord in dense graphs, Discret. Math., 341, 2131-2141, (2018) · Zbl 1388.05104 [4] Cream, M.; Gould, RJ; Hirohata, K., A note on extending Bondy’s meta-conjecture, Australas. J. Combin., 67, 463-469, (2017) · Zbl 1375.05073 [5] Faudree, RJ; Gould, RJ; Jacobson, MS, Pancyclic graphs and linear forests, Discret. Math., 309, 1178-1189, (2009) · Zbl 1173.05026 [6] Faudree, RJ; Gould, RJ; Jacobson, MS; Lesniak, L., Generalizing pancyclic and $$k$$-ordered graphs, Graphs Combin., 20, 291-309, (2004) · Zbl 1056.05086 [7] Finkel, D., On the number of independent chorded cycles in a graph, Discret. Math., 308, 5265-5268, (2008) · Zbl 1228.05170 [8] Gould, R.J.: Graph Theory. Dover Publications Inc., Mineola (2012) · Zbl 1284.05003 [9] Gould, RJ; Hirohata, K.; Horn, P., On independent doubly chorded cycles, Discret. Math., 338, 2051-2071, (2015) · Zbl 1314.05103 [10] Hendry, GRT, Extending cycles in graphs, Discret. Math., 85, 59-72, (1990) · Zbl 0714.05038 [11] Ore, O., Note on Hamilton circuits, Am. Math. Mon., 67, 55, (1960) · Zbl 0089.39505 [12] Randerath, B.; Schiermeyer, I.; Tewes, M.; Volkmann, L., Vertex pancyclic graphs, Discret. Appl. Math., 120, 219-237, (2002) · Zbl 1001.05070
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