A known result which plays an importanl role in claw-free hamiltonian graph theory is that every 2-connected claw-free graph on n venices contains a cycle of length at least min { 2δ + 4.n }

and is hamiltonian of n ≤3δ+ 2. In this paper

we generalile the result and show that every 2-connected claw-free graph G D contains a cycle of length at leant mind {3δ+2

n }

where D is the set of all the graphs defined as follows: Any graph G of order at most (9δ)/2 - 1 in D can be decomposed into three disjoint hamiltonian subgraphs C1 .G2 and C3 such that EG (G.

Gj ) = {u.u

.viv } for i ≠ j and i.j =1

2

3 (where ). ∈ V(G1) for

i= 1. 2 .3 ) and at most one subgraph C

has 2δ-2 venices. As the corollary of our result

we still obtain that every 2-connected clawfree graph G D on n ≤ 4δ venices is hamiltonian. Moreover the bounds 3δ+2 and 4δare bestpossible.