6 with For the infinite-length strip of the square lattice with width 4 and free transverse boundary conditions, sq [4$_{F}$ × ∞ ] [31, 35] the fact that the effective coordination number increases monotonically as a function of L$_{z}$ . The use of periodic boundary conditions in the z direc- tion minimizes finite-size effects, so that for a given L$_{z}$ , W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q ) would be expected to be closer to W ( sc, q ) than W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q ) [30]. Again, to the extent that the lower bounds are close to the actual W functions for these respective slabs, one would expect W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q )$_{ℓ}$ to be closer than W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q )$_{ℓ}$ to W ( sc, q ). Our results agree with this expectation. In contrast to W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q ), W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q ) is not, in general, a non-increasing function of L$_{z}$ , as was discussed in general in [30] (see Fig. 1 therein). Thus, values of W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q ), and hence, a for- tiori , W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q )$_{ℓ}$, may actually lie slightly below those for W ( sc, q ), as is evident for the W ( S$_{3}$$_{P}$ , q )$_{ℓ}$ entries in Table I. where λ$_{4}$$_{F}$$_{,max}$ is the largest root of the cubic equation with and R$_{3}$ = ( q 2 − 5 q + 7)( q 4 − 5 q 3 + 11 q 2 − 12 q + 8) . (9.3) W ( sq [4$_{F}$ × ∞ ] , q ) = ( λ$_{4}$$_{F}$$_{,max}$ ) 1 / 4 (9.4) x 3 + b$_{4}$$_{F}$$_{,}$$_{1}$x 2 + b$_{4}$$_{F}$$_{,}$$_{2}$x + b$_{4}$$_{F}$$_{,}$$_{3}$ = 0 (9.5) b$_{4}$$_{F}$$_{,}$$_{1}$ = − q 4 + 7 q 3 − 23 q 2 + 41 q − 33 (9.6) b$_{4}$$_{F}$$_{,}$$_{2}$ = 2 q 6 − 23 q 5 + 116 q 4 − 329 q 3 + 553 q 2 − 517 q + 207 (9.7) b$_{4}$$_{F}$$_{,}$$_{3}$ = − q 8 + 16 q 7 − 112 q 6 + 449 q 5 − 1130 q 4 + 1829 q 3 − 1858 q 2 + 1084 q − 279 . (9.8) In this paper we have calculated rigorous lower bounds for the ground state degeneracy per site W , equivalent to the ground state entropy S$_{0}$ = k$_{B}$ ln W , of the q -state Potts antiferromagnet on slabs of the simple cubic lattice that are infinite in two directions and finite in the third. Via comparison with large- q expansions and numerical evaluations, we have shown how the results interpolate between the square (sq) and simple cubic (sc) lattices. Acknowedgments: This research was supported in part by the NSF grant PHY-06-53342. RS expresses his grat- itude to coauthors on previous related works, in partic- ular, S.-H. Tsai and J. Salas, as well as N. Biggs, S.-C. Chang, and M. Roˇ cek. VIII. CONCLUSIONS IX. APPENDIX We note the following results on E d lattices and lat- tice sections: W (Λ$_{bip.}$ , 2) = 1 for any bipartite lattice; W ( sq, 3) = (4 / 3) 3 / 2 [37]; and W ( { L } , q ) = W ( { C } , q ) = q − 1, where L$_{n}$ and C$_{n}$ denote the n -vertex line and cir- cuit graphs. For the infinite-length square-lattice strip of width 2, W ( sq [2$_{F}$ × ∞ ] , q ) = W ( sq [2$_{P}$ × ∞ ] , q ) = √ q 2 − 3 q + 3, where, as in the text, the subscripts F and P denote free and periodic boundary conditions in the direction in which the strip is finite. For the infinite- length strip of the square lattice with (transverse) width 3 and free transverse boundary conditions, sq [3$_{F}$ × ∞ ] [31–33] where W ( sq [3$_{F}$ × ∞ ] , q ) = ( λ$_{3}$$_{F}$$_{,max}$ ) 1 / 3 (9.1) λ$_{3}$$_{F}$$_{,max}$ = 1 2 [ ( q − 2)( q 2 − 3 q + 5) + √ R$_{3}$ ] (9.2)