Constructions of bi-regular cages

Given three positive integers r,m and g, one interesting question is the following: What is the minimum number of vertices that a graph with prescribed degree set {r,m}, 2 ≤ r < m, and girth g can have? Such a graph is called a bi-regular cage or an ({r,m}; g)-cage, and its minimum order is denoted by n({r,m}; g). In this paper we provide new upper bounds on n({r,m}; g) for some related values of r and m. Moreover, if r − 1 is a prime power, we construct the following bi-regular cages: ({r, k(r − 1)}; g)-cages for g ∈ {5, 7, 11} and k ≥ 2 even; and ({r, kr}; 6)-cages for k ≥ 2 any integer. The latter cages are of order n({r, kr}; 6) = 2(kr − kr+1). Then this result supports the conjecture that n({r,m}; 6) = 2(rm−m+1) for any r < m, posed by Yuansheng and Liang [The minimum number of vertices with girth 6 and degree set D = {r,m}, Discrete Mathematics 269 (2003), 249-258]. We finalize giving the exact value n({3, 3k}; 8), for k ≥ 2. keywords : cage graphs, girth This work has been published in Discrete Maths. 309(6) (2009) 1409–1416. Corresponding author, supported by PAPIIT-UNAM, México, Proyecto IN106305-3 This research was supported by the Ministry of Science and Innovation, Spain, and the European Regional Development Fund (ERDF) under project MTM2008-06620-C03-02/MTM