Exact value of ex(n;{C3,,Cs}) e x ( n ; { C 3 ,  , C s } ) for n ? ? 25 ( s ? 1 ) 8 ?
For integers s?8 s ? 8 and s + 1 ? n ? ? 25 ( s ? 1 ) 8 ? , we determine the exact value of the function ex(n;{C3,,Cs}) e x ( n ; { C 3 ,  , C s } ) , that represents the maximum number of edges in a {C3,,Cs} { C 3 ,  , C s } -free graph of order n n . This result was already known when 3?s?7 3 ? s ? 7 . To do that, for 1?k?5 1 ? k ? 5 , we provide a family of graphs H s k such that e ( H s k ) ? n ( H s k ) = k and with the property that H s k reaches girth s+1 s + 1 with the minimum number of vertices. Also, we determine an infinity family of solutions of the problem ex(n;{C3,,Cs})=n+6 e x ( n ; { C 3 ,  , C s } ) = n + 6 .
girth extremal function extremal graphs forbidden cycles