Abstract
Let r, m, 2 ≤ r < m and g ≥ 3 be three positive integers. A graph with a prescribed degree set r, m and girth g having the least possible number of vertices is called a bi-regular cage or an (r, m; g)-cage, and its order is denoted by n(r, m; g). In this paper we provide upper bounds on n(r, m; g) for some related values of r, m and even girth g at least 8. Moreover, if r − 1 is a prime power and m ≥ 5, we construct the smallest currently known (r, m; 8)-graphs. Also, if r = 3 and m ≥ 7 is not divisible by 3, we prove that \({n(3,m; 8) \ge \lceil 25m/3\rceil + 7}\). Finally, we construct a family of (3, m; 8)-graphs of order 9m + 3 which are cages for m = 4,5,7.
Published Version
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