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if no conditions are imposed, an obvious upper bound on the number of proper q -colorings of the vertices of G is P ( G, q ) ≤ q $^{n}$. This yields the corresponding upper bound W ( { G } , q ) < q . Hence, it is natural to define a reduced function that has a finite limit as q → ∞ , 

For a lattice or, more generally, a graph whose vertices have bounded degree, W$_{r}$ ( { G } , q ) is analytic about 1 /q = 0. ( W$_{r}$ ( { G } , q ) is non-analytic at 1 /q = 0 for certain families of graphs that contain one or more vertices with unbounded degree as n → ∞ , although the presence of a vertex with unbounded degree in this limit does not necessarily imply non-analyticity of W$_{r}$ ( { G } , q ) at 1 /q = 0 [21, 22].) It is conventional to express the large- q Taylor series for a function that has some factors removed from W$_{r}$ , since this function yields a simpler expansion. A chromatic polynomial has the general form 

where the a$_{n}$ $^{−}$j > 0 and k ( G ) is the number of connected components of G (taken here to be k ( G ) = 1 without loss of generality). One has a$_{n}$ = 1, a$_{n}$ $^{−}$1 = e ( G ), and, provided that the girth g ( G ) > 3 [23], as is the case here, a$_{n}$ $^{−}$2 = ( e ( G ) 2 ) . A κ -regular graph is a graph such that each vertex has degree (coordination number) κ . For a κ -regular graph, e ( G ) = κn/ 2. The coefficients of the three terms of highest degree in q in P ( G, q ) for a κ - regular graph are precisely the terms that would result from the expansion of [ q (1 − q $^{$^{−}$1}$) κ/ $^{2}$] $^{n}$. Hence, for a κ - regular graph or lattice, one usually displays the large- q series expansions for the reduced function 

The large- q Taylor series for this function can be written in the form 

where 

The two results that we shall need here are the large- q (i.e., small- y ) Taylor series for W ( sq, q ) and W ( sc, q ). The large- q series for W ( sq, q ) was calculated to succes- sively higher orders in [24]-[28]. Here we only quote the terms to O ( y $^{11}$): 

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As noted above, lower bounds on W (Λ , q ) obtained from the inequality (2.12) for two-dimensional lattices Λ were found to be quite close to the actual values of the respective W (Λ , q ) for a large range of values of q . This can be understood for large values of q from the fact that they coincide with the large- q expansions to many orders, and the agreement actually extends to values of q only moderately above q = 2. For example, the lower bound on W ( sq, q ) in [11] is equivalent to W ( sq, q ) ≥ (1 + y $^{3}$). This agrees with the small- y series up to order O ( y $^{6}$), as is evident from comparison with Eq. (4.6). This lower bound also agrees quite closely with the value of W ( sq, q ) determined by Monte Carlo simulations in [10, 12, 13] (see Table 1 of [10] and Table 1 of [12]). We include this comparison here in Table I. For our purposes, it is sufficient to quote the results from Ref. [10] only to three significant figures. Since we are using large- q series for this comparison, we list the results in Table I for a set of values q ≥ 4. As another example, the lower bound obtained for the honeycomb lattice in Ref. [13], W ( hc, q ) ≥ (1 + y $^{5}$)1 / $^{2}$, agreed with the small- y series for W ( hc, q ) to O ( y $^{10}$). Thus, it was found that for all of the cases studied, W (Λ , q )$_{ℓ}$ provides not only a lower bound on W (Λ , q ), but a rather good approximation to the latter function. It is thus reasonable to expect that this will also be true for the lower bounds W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{BC}$$_{z}$ , q )$_{ℓ}$ for the slabs S$_{L}$$_{z}$ of the simple cubic lattice considered here, of infinite extent in the x and y directions and of thickness L$_{z}$ in the z direction. 

From ingredients given in Ref. [25], we have calculated a large- q expansion of the W ( sc, q ) for the simple cubic ( sc ) lattice and obtain 

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In Table I we list the corresponding values of W ( sc, q ) obtained from this large- q series, denoted W ( sc, q )$_{ser.}$, for q ≥ 4. We also list estimates of W ( sc, q ), denoted W ( sc, q )$_{MC}$, for 4 ≤ q ≤ 6 from the Monte Carlo calcu- lations in Ref. [29]. One sees that the approximate values obtained from the large- q series are close to the estimates from Monte Carlo simulations even for q values as low as q = 4. 

The coordination number for the S$_{2}$$_{F}$ slab of the sim- ple cubic lattice (of infinite extent in the x and y direc- tions) is κ ( S$_{2}$$_{F}$ ) = 5. We thus analyze the reduced func- tion W ( S$_{2}$, q )$_{ℓ}$ = W ( S$_{2}$, q )$_{ℓ}$ / [ q (1 − q $^{$^{−}$1}$) 5 / $^{2}$]. This has the large- q (small- y ) expansion 

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As this shows, W ( S$_{2}$, q )$_{ℓ}$ provides an interpolation be- tween W ( sq, q ) and W ( sc, q ); for example, the coefficient of the y 3 term is 1 for Λ = sq , 2 for Λ = S$_{2}$ , and 3 for Λ = sc . Furthermore, the coefficient of the y 5 term is 0 for Λ = sq , 2 for Λ = S$_{2}$ , and 22 for Λ = sc . This is in agreement with the fact that the exact functions W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q ) interpolate between W ( sc, q ) and W ( sc, q ) as L$_{z}$ increases from 1 to ∞ [30] and the expectation, 

