Graphs with maximum size and lower bounded girth

AbstractFor integers n≥4 and ν≥n+1, let ex(ν;{C3,…,Cn}) denote the maximum number of edges in a graph of order ν and girth at least n+1. The {C3,…,Cn}-free graphs with order ν and size ex(ν;{C3,…,Cn}) are called extremal graphs and denoted by EX(ν;{C3,…,Cn}). We prove that given an integer k≥0, for each n≥2log2(k+2) there exist extremal graphs with ν vertices, ν+k edges and minimum degree 1 or 2. Considering this idea we construct four infinite families of extremal graphs. We also see that minimal (r;g)-cages are the exclusive elements in EX(ν0(r,g);{C3,…,Cg−1})