Girth of {C3,,Cs} { C 3 ,  , C s } -free extremal graphs ?
LetG G be a {C3,,Cs} { C 3 ,  , C s } -free graph with as many edges as possible. In this paper we consider a question studied by several authors, the compulsory existence of the cycleCs+1 C s + 1 inG G . The answer has already been proved to be affirmative for s=3,4,5,6 s = 3 , 4 , 5 , 6 . In this work we show that the girth of G G is g(G)=s+1 g ( G ) = s + 1 when the order ofG G is at least 1 + s ( s ? 2 2 ) s ? 2 ? 4 s ? 4 ifs s is even, and 1 + ( s ? 1 ) 3 ( ( s ? 2 ) 2 ? 1 4 ) s ? 3 2 ? 8 s 2 ( s ? 2 ) 2 ? 10 ifs s is odd. This bound is an improvement of the best general result so far known. Moreover, we also prove in the case s=7 s = 7 that the girth is g(G)=8 g ( G ) = 8 for order at least 14 and characterize all the extremal graphs whose girth is not8 8 .
girth extremal graph extremal function degree of an edge