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

Abstract

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.