5 One then needs λ$_{tube}$$_{(3}$$_{P}$$_{)}$$_{,max}$ . The relevant (4 × 4 di- mensional) transfer matrix for this tube graph was cal- culated for Ref. [34], and we have used this to compute λ$_{tube}$$_{(3}$$_{P}$$_{)}$$_{,max}$ numerically. The results for W ( S$_{3}$$_{P}$ , q ) are given in Table I. as discussed above, that W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q )$_{ℓ}$ should be close to W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q ). Since the slabs of infinite extent in the x and y direc- tions and of finite thickness L$_{z}$ geometrically interpolate between the square and simple cubic lattices, it follows that the resultant W functions for these slabs interpo- late between W ( sq, q ) and W ( sc, q ) [30]. Given that it was shown previously that the lower bounds W (Λ , q )$_{ℓ}$ obtained by the coloring matrix method are quite close to the actual values of the respective W (Λ , q ) for a num- ber of two-dimensional lattices, this is also expected to be true for the W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{BC}$$_{z}$ , q )$_{ℓ}$ bounds. We have shown above how W ( S$_{2}$, q )$_{ℓ}$ interpolates between W ( sq, q ) and W ( sc, q ) via a comparison of the large- q series expan- sions for these three functions. Table I provides a fur- ther numerical comparison for W ( S$_{3}$$_{F}$ , q )$_{ℓ}$, W ( S$_{4}$$_{F}$ , q ), and W ( S$_{3}$$_{P}$ , q ) with W ( sq, q ) and W ( sc, q ), the latter be- ing determined to reasonably good accuracy from large- q series expansions and, where available, Monte Carlo measurements. As noted, the lower end of the range of q values for the comparison is chosen as q = 4 in view of the use of large- q series. For the slab of the simple cubic lattice with thick- ness L$_{z}$ = 3 and FBC$_{z}$, denoted S$_{3}$$_{F}$ , the transverse slice is the graph sq [3$_{F}$ × ( L$_{y}$ )$_{BCy}$]. The chromatic poly- nomials P ( sq [3$_{F}$ × ( L$_{y}$ )$_{F}$ ] , q ), P ( sq [3$_{F}$ × ( L$_{y}$ )$_{P}$ ] , q ), and P ( sq [3$_{F}$ × ( L$_{y}$ )$_{TP}$ ] , q ) (where TP denotes twisted peri- odic, i.e., M¨ obius BC) were computed for arbitrary L$_{y}$ in Refs. [31], [32], and [33], respectively, and the maxi- mal λ was shown to be the same for all of these bound- ary conditions. For the reader’s convenience, we list this λ$_{ts}$$_{(3}$$_{F}$$_{)}$$_{,max}$ in Eqs. (9.2) and (9.3) of the Appendix. The other input that is needed to obtain the lower bound in Eq. (2.14) is the maximal λ for the chromatic polynomial of the sc [2$_{F}$ × 3$_{F}$ × L$_{y}$ ] tube graph, i.e., λ$_{tube}$$_{(3}$$_{F}$$_{)}$$_{,max}$ . The relevant transfer matrix that determines the chromatic polynomial for this tube graph was given with Ref. [34]. Because it is 13 × 13 dimensional, one cannot solve the corresponding characteristic polynomial analytically to obtain λ$_{tube}$$_{(3}$$_{F}$$_{)}$$_{,max}$ for general q . However, one can cal- culate λ$_{tube}$$_{(3}$$_{F}$$_{)}$$_{,max}$ numerically, and we have done this. Combining these results with Eqs. (9.2) and (9.3), we then evaluate the lower bound W ( S$_{3}$$_{F}$ , q )$_{ℓ}$ by evaluating the L$_{z}$ = 3 special case of (2.14). VII. DISCUSSION V. RESULTS FOR SLABS OF THICKNESS L z = 3 , 4 WITH FBC$_{z}$ VI. RESULT FOR SLAB OF THICKNESS L z = 3 WITH PBC$_{z}$ For sections of lattices, and, more generally, graphs that are not κ -regular, one can define an effective vertex degree (coordination number) as [14] For the slab of the simple cubic lattice with thickness L$_{z}$ = 4 and FBC$_{z}$, S$_{4}$$_{F}$ , the transverse slice is the graph sq [4$_{F}$ × ( L$_{y}$ )$_{BCy}$]. Here the maximal λ$_{ts}$$_{(4}$$_{F}$$_{)}$$_{,max}$ is the so- lution of the cubic equation (9.5) given in the Appendix. One also needs the maximal λ for the chromatic polyno- mial of the sc [2$_{F}$ × 4$_{F}$ × L$_{y}$ ] tube graph, i.e., λ$_{tube}$$_{(4}$$_{F}$$_{)}$$_{,max}$ . The relevant (136 136 dimensional) transfer matrix for this tube graph was calculated for Ref. [34], and we have used this to compute λ$_{tube}$$_{(4}$$_{F}$$_{)}$$_{,max}$ numerically. We then obtain the lower bound W ( S$_{4}$$_{F}$ , q )$_{ℓ}$ from the L$_{z}$ = 4 spe- cial case of Eq. (2.14). The results for W ( S$_{3}$$_{F}$ , q ) and W ( S$_{4}$$_{F}$ , q ) are listed in Table I. For 3 ≤ L$_{z}$ < ∞ , the slab of the simple cubic lattice (of infinite extent in the x and y directions) with FBC$_{z}$ is not κ -regular, but has the effective coordination number It is also of interest to obtain a lower bound for W for a slab with periodic boundary conditions in the z direction, since these minimize finite-volume effects. For this purpose we consider the slab of the simple cubic lattice with thickness L$_{z}$ = 3 and PBC$_{z}$, S$_{3}$$_{P}$ . In this case the transverse slice is the graph sq [3$_{P}$ × ( L$_{y}$ )$_{BCy}$]. For FBC$_{y}$ the chromatic polynomial involves only one λ , and this is also the maximal λ for PBC$_{y}$ and TPBC$_{y}$ [36], viz., κ$_{eff}$ = 2 e ( G ) n ( G ) . (7.1) κ$_{eff}$ ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ ) = 2 ( 3 − 1 L$_{z}$ ) . (7.2) λ$_{ts}$$_{(3}$$_{P}$$_{)}$$_{,max}$ = q 3 − 6 q 2 + 14 q − 13 . (6.1) We observe that for the q values considered in Table I, W ( sq, q ) > W ( S$_{2}$, q )$_{ℓ}$ > W ( S$_{3}$$_{F}$ , q )$_{ℓ}$ > W ( S$_{4}$$_{F}$ , q ) > W ( sc, q ). The fact that for fixed q , the exact function W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q ) is a non-increasing function of L$_{z}$ , and, for q > 2, a monotonically decreasing function of L$_{z}$ , follows from a theorem proved in Ref. [30]. To the extent that the lower bounds W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q )$_{ℓ}$ lie close to the actual val- ues of W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q ), it is understandable that they also exhibit the same strict monotonicity. As was noted in Ref. [30], the reason for the monotonicity of the exact values is that the number of proper q -colorings per ver- tex of a lattice graph is more highly constrained as one increases the effective coordination number of the lattice section. (This is also evident in Fig. 5 of [10].) In the present case, the monotonicity can be seen as a result of