Gao (Z.). -- A pattern for the asymptotic number of rooted maps on surfaces. Journal of Combinatorial Theory. Series A, vol. 64, n 2, 1993, pp. 246--264.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....g 0 . At the end of the game, elements in A are either too far away to interact or share the same anchor. In both cases the corresponding elements in B will satisfy the same atomic formulas. 6. Conclusion Richness properties have been proved for many families of rooted maps by Bender et al. [4, 3, 5, 6, 7]. Thanks to the work of Richmond and Wormald [8] these results hold as well for unrooted maps. Using the previously described machinery it implies 0 1 laws for the following families of maps on a surface of given genus: all maps, smooth maps, 2 connected maps, 3 connected maps, triangular maps, ....

Gao (Z.). -- A pattern for the asymptotic number of rooted maps on surfaces. Journal of Combinatorial Theory. Series A, vol. 64, n 2, 1993, pp. 246--264.

.... Large representativity for (i) ii) and (iii) follows from [5] Large representativity for (iv) follows from [5] and [7] Large representativity for (v) follows from [5] and [20] Large representativity for (vi) follows from [5] and [21] Large representativity for (vii) follows from [5] and [22]. The results in these references are all stated for rooted maps. However, by [29] they are valid as stated here. Remark. For maps on the plane, large representativity is trivial and so only richness needs to be verified for a 0 1 law to hold. Richness has been proved for many classes of planar ....

Zhi-Cheng Gao. A pattern for the asymptotic number of rooted maps on surfaces. J. Combin. Theory Ser. A, 64:246--264, 1992.

....a formula or convince your colleagues that one probably does not exist. There may well be no simple formula. Arqu es and Giorgetti [1] may have done as much as possible. 4. Asymptotic Problems A variety of classes of maps have been studied asymptotically in surface of arbitrary genus. See [13]. Of course, one can always ask for the asymptotics of another class of maps in arbitrary surfaces. I ll focus on some more general questions. One would expect almost all maps in almost any class to be asymmetric. If this were true, it would lead to a simple asymptotic relationship between maps ....

....been enumerated asymptotically on arbitrary surfaces, MS (n) ff(S; C)n Gamma5(S) 4 fi(C) n ; 4:1) where (S) is the Euler characteristic of S and fi is independent of S. In addition, Gao has discovered that ff has a remarkable structure. For a fuller discussion of the pattern in (4. 1) see [13]. Of course, the possibility always exists that this pattern is an artifact: those maps which were amenable to study have a particular structure to their generating functions which makes them amenable. I believe that the pattern holds much more generally than has been established. One of the ....

Z.C. Gao, A pattern for the asymptotic number of rooted maps on surfaces, J. Combin. Theory Ser. A 64 (1993) 246--264.

....rooted maps. Proposition 1. Let D be a finite subset of the positive integers. The normal growth of the following families of rooted maps is proved in the locations cited. 1. rooted maps [1] 2. rooted smooth maps [1] 3. rooted loopless maps, rooted simple maps and rooted 3 c triangulations [10]; 4. rooted nonsingular maps and rooted 2 c maps (and the numbers are asymptotically equal) 5] 4 5. rooted triangular maps [7] 6. rooted triangulations, that is, rooted 2 c triangular maps [8] 7. for certain D as discussed in [9] rooted maps with all face valencies in D (D = f3g is case ....

Z. Gao, A pattern for the asymptotic number of rooted maps on surfaces, J. Combin. Theory, Ser. A, to appear.

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