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We prove that a weighted Coxeter group (W, S, L) is bounded with a(W ) = b′(W ) := max{L(u), L(ws,t) | u, s, t ∈ S, |Ws,t| < ∞} if the Coxeter graph of W is complete and b′(W ) < ∞, where Ws,t is the parabolic subgroup of W generated by s 6= t in S and ws,t is the longest element in Ws,t whenever Ws,t is finite.. Let W be a Coxeter group with S its Coxeter generator set and ` its length function. In [2], Lusztig defined a weight function L on (W,S). Call (W,S,L) a weighted Coxeter group. Let A = Z[v, v−1] be the ring of Laurent polynomials in an indeterminate v with integer coefficients. In [2], Lusztig introduced an A-basis {cw | w ∈ W} in the IwahoriHecke algebra H associated to (W,S,L). For any x, y, z ∈ W , define hx,y,z ∈ A by the expression cxcy = ∑ z∈W hx,y,zcz. Call (W,S,L) bounded if a(W ) := max{deg hx,y,z | x, y, z ∈ W} < ∞. In [6, Theorem 2.1], Xi asserted that (W,S, `) is bounded if its Coxeter graph Γ(W ) is complete and the cardinalities of its finite parabolic subgroups have a common upper bound. In [4], we proved that the weighted universal Coxeter group is bounded if the set L(S) := {L(s) | s ∈ S} is finite. For any s 6= t in S, denote by Ws,t the subgroup of W generated by s, t and by wst the longest element in Ws,t whenever Ws,t is finite. In this paper, we shall prove in Theorem 3.2 that a weighted Coxeter group (W,S,L) is bounded if (W,S,L) satisfies the condition (∗) below: (∗) Γ(W ) is complete and b′(W ) := max{L(u), L(wst) | u, s, t ∈ S, |Ws,t| < ∞} < ∞,

The boundedness of the weighted Coxeter group with complete graph