On the order of bi-regular cages of even girth

By a bi-regular cage of girth g we mean a graph with prescribed degrees r and m and with the least possible number of vertices denoted by f ( { r , m } ; g ) . We provide new upper and lower bounds of f ( { r , m } ; g ) for even girth g ⩾ 6 . Moreover, we prove that f ( { r , k ( r − 1 ) + 1 } ; 6 ) = 2 k ( r − 1 ) 2 + 2 r where k ⩾ 2 is any integer and r − 1 is a prime power. This result supports the conjecture f ( { r , m } ; 6 ) = 2 ( r m − m + 1 ) for any r < m formulated by Yuansheng and Liang (The minimum number of vertices with girth 6 and degree set D = { r , m } , Discrete Mathematics 269 (2003), 249–258).