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Some properties of semiregular cages

Abstract : A graph with degree set \r, r + 1\ is said to be semiregular. A semiregular cage is a semiregular graph with given girth g and the least possible order. First, an upper bound on the diameter of semiregular graphs with girth g and order close enough to the minimum possible value is given in this work. As a consequence, these graphs are proved to be maximally connected when the girth g >= 7 is odd. Moreover an upper bound for the order of semiregular cages is given and, as an application, every semiregular cage with degree set \r, r + 1\ is proved to be maximally connected for g is an element of \6, 8\, and when g = 12 for r >= 7 and r not equal 20. Finally it is also shown that every (\r, r + 1\; g)-cage is 3-connected.
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Camino Balbuena, Xavier Marcote, Diego Gonzalez-Moreno. Some properties of semiregular cages. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2010, 12 (5), pp.125-138. ⟨hal-00990460⟩

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