Cycles in 2-connected graphs

Abstract

Let G n be a class of graphs on n vertices. For an integer c, let ex ( G n , c ) be the smallest integer such that if G is a graph in G n with more than ex ( G n , c ) edges, then G contains a cycle of length more than c. A classical result of Erdös and Gallai is that if G n is the class of all simple graphs on n vertices, then ex ( G n , c ) = c 2 ( n - 1 ) . The result is best possible when n - 1 is divisible by c - 1 , in view of the graph consisting of copies of K c all having exactly one vertex in common. Woodall improved the result by giving best possible bounds for the remaining cases when n - 1 is not divisible by c - 1 , and conjectured that if G n is the class of all 2-connected simple graphs on n vertices, then ex ( G n , c ) = max { f ( n , 2 , c ) , f ( n , ⌊ c / 2 ⌋ , c ) } , where f ( n , t , c ) = c + 1 - t 2 + t ( n - c - 1 + t ) , 2 ⩽ t ⩽ c / 2 , is the number of edges in the graph obtained from K c + 1 - t by adding n - ( c + 1 - t ) isolated vertices each joined to the same t vertices of K c + 1 - t . By using a result of Woodall together with an edge-switching technique, we confirm Woodall's conjecture in this paper.