Cycles in 2-connected graphs

نویسندگان

  • Genghua Fan
  • Xuezheng Lv
  • Pei Wang
چکیده

Let Gn be a class of graphs on n vertices. For an integer c, let ex(Gn, c) be the smallest integer such that if G is a graph in Gn with more than ex(Gn, c) edges, then G contains a cycle of length more than c. A classical result of Erdös and Gallai is that if Gn is the class of all simple graphs on n vertices, then ex(Gn, 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 Kc 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 Gn is the class of all 2-connected simple graphs on n vertices, then ex(Gn, 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 Kc+1−t by adding n− (c+ 1− t) isolated vertices each joined to the same t vertices of Kc+1−t . By using a result of Woodall together with an edge-switching technique, we confirm Woodall’s conjecture in this paper. © 2004 Elsevier Inc. All rights reserved.

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عنوان ژورنال:
  • J. Comb. Theory, Ser. B

دوره 92  شماره 

صفحات  -

تاریخ انتشار 2004