Geodesic
Hamiltonian and Pancyclic Graphs in the Class of Self-Centered Graphs with Radius Two
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 3, pp. 661-681

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The paper deals with Hamiltonian and pancyclic graphs in the class of all self-centered graphs of radius 2. For both of the two considered classes of graphs we have done the following. For a given number n of vertices, we have found an upper bound of the minimum size of such graphs. For n ≤ 12 we have found the exact values of the minimum size. On the other hand, the exact value of the maximum size has been found for every n. Moreover, we have shown that such a graph (of order n and) of size m exists for every m between the minimum and the maximum size. For n ≤ 10 we have found all nonisomorphic graphs of the minimum size, and for n = 11 only for Hamiltonian graphs.
Keywords: self-centered graph with radius 2, Hamiltonian graph, pancyclic graph, size of graph 
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Hrnčiar, Pavel; Monoszová, Gabriela. Hamiltonian and Pancyclic Graphs in the Class of Self-Centered Graphs with Radius Two. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 3, pp. 661-681. http://geodesic.mathdoc.fr/item/DMGT_2018_38_3_a4/