The minimum number of vertices with girth 6 and degree set D = {r,m}

A (D;g)-cage is a graph having the minimum number of vertices, with degree set D and girth g. Denote by f(D;g) the number of vertices in a (D;g)-cage. In this paper it is shown that f({r,m};6)≥2(rm-m+1) for any 2≤r<m, and f({r,m};6)=2(rm-m+1) if either (i) 2≤r≤5 and r<m or (ii) m-1 is a prime power and 2≤r<m. Upon these results, it is conjectured that f({r,m};6)=2(rm-m+1) for any r with 2≤r<m.