If G is a finitely presented group and [Kscr] is any (G,2)-complex (that is, a finite 2-complex with fundamental group G), then it is well known that χ([Kscr]) [ges ] ν(G), where ν(G) = 1−rk H1G+dH2G. We define χ(G) to be min{χ([Kscr]): [Kscr] a (G, 2)-complex}, and we say that G is efficient if χ(G) = ν(G). In this paper we give sufficient conditions for a Coxeter group to be efficient (Theorem 4.2). We also give examples of inefficient Coxeter groups (Theorem 5.1). In fact, we give an infinite family Gn(n = 2, 3, 4, . . . ) of Coxeter groups such that χ(Gn)−ν(Gn) [xrarr ] ∞ as n [xrarr ] ∞.