Size of Graphs with High Girth

Let n ⩾ 4 be a positive integer and let e x ( ν ; { C 3 , … , C n } ) denote the maximum number of edges in a { C 3 , … , C n } -free simple graph of order ν. This paper gives the exact value of this function for all ν up to ⌊ ( 16 n − 15 ) / 5 ⌋ . This result allows us to deduce all the different values of the girths that such extremal graphs can have. ⩾ 0 be an integer. For each n ⩾ 2 log 2 ( k + 2 ) there exists ν such that every extremal graph G with e ( G ) − ν ( G ) = k has minimal degree at most 2, and is obtained by adding vertices of degree 1 and/or by subdividing a graph or a multigraph H with δ ( H ) ⩾ 3 and e ( H ) − ν ( H ) = k .