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)=2k2(r−1)+2r where k⩾2 is any integer and r−1 is a prime power. This result supports the conjecture f({r,m};6)=2(rm−m+1) for any r