Enumeration of Polyomino Tilings via Hypergraphs

Transcription

1 Enumeration of Polyomino Tilings via Hypergraphs (Dedicated to Professor Károly Bezdek) Muhammad Ali Khan Centre for Computational and Discrete Geometry Department of Mathematics & Statistics, University of Calgary S, February 14, 2015

2 Finite tiling Problem Given a finite region R consisting of (not necessarily connected) cells from a regular lattice, how many ways are there to tile R by a finite set of tiles, where each tile is a union of (not necessarily connected) lattice cells? For example, tiling a rectangle by polyominoes, tiling a finite portion of the hexagonal lattice by polyhexes.

3 Some solved rectangular tilings Polyomino References Domino Fisher and Temperley (1961), Kasteleyn (1961) L-tromino Chinn, Grimaldi and Heubach (2007) (for 2 n and 3 n strips) T-tetromino Korn and Pak (2004), Jacobsen (2007) (connections with Tutte polynomial) Generalized T-tetromino Kayibi and Pirzada (2012) (connections with Tutte polynomial)

4 Automatic CounTilings 1. D. Zeilberger, Automatic countilings, (2006). Description of a Maple program that takes the dimensions of a rectangular grid and a set of tiles as input and outputs the number of tilings of the grid by the given tiles. Examples are provided but not the formal proof that the program always works. 2. D. Merlini, R. Sprugnoli, M. C. Verri, Strip tiling and regular grammars, Theoretical Computer Science 242 (2000), Proves that every tiling of an m n rectangle by polyominoes is equivalent to a regular grammar. Proposes an algorithm to transform the tiling problem into a grammar and count the tilings as the number of words of length n in the corresponding language. Benedetto and Loehr (2008) extended this approach to other lattices and lattice animals.

5 The edge cover polynomial An edge covering of a graph G(V, E) is a set of edges such that every vertex of the graph is incident to at least one edge of the set. Akbari and Oboudi (2013) introduced the edge cover polynomial E(G, x) of a graph G as the generating function of the number of edge coverings of G. That is E E(G, x) = e(g, i)x i, i=1 where e(g, i) is the number of edge covers of G of size i. They studied some algebraic and combinatorial properties of the edge cover polynomial and gave a recursive procedure for determining E(G, x) for any simple graph G.

6 Edge cover polynomial of hypergraphs Let H(V, E) be a simple (not necessarily uniform) hypergraph. We define its edge coverings analogously and its edge cover polynomial as C(H, x) = m c(h, i)x i, i=1 where c(h, i) is the number of edge coverings of H of size i. In fact, if ρ(h) denotes the edge covering number of H, that is the size of a minimum edge covering of H, then C(H, x) = E i=ρ(h) c(h, i)x i.

7 A deletion-based recursion Let us denote the operation of deleting a vertex v (and the edges containing v) from a hypergraph H as H v, the operation of deleting the a set U of vertices (and the edges containing these vertices) from H as H U, and the operation of deleting an edge e from H as H e. Theorem 1 Let H(V, E) be a simple hypergraph with rank at least 2 and e be an edge of H, with V (e) denoting the set of vertices of e, then C(H, x) = (x + 1)C(H e, x) + x U V (e) C(H U, x). (1)

8 Proof Let δ(h) denote the minimum degree of a vertex in H and for any vertex v of H let N(v) = {e E : v e}. If δ(h) = 0, then there is nothing to prove. So WLOG, assume that δ(h) 1. Let S be an edge covering of H with size i. 1 If e / S, then S is an edge covering of size i for H e. 2 If e S, let We have the following cases: (a) U = : U = {v V (e) : S N(v) = 1}. S\{e} is an edge covering of size i 1 of H e. (b) U : S\{e} is an edge covering of size i 1 of H U.

9 Proof Therefore, c(h, i) = c(h e, i) + c(h e, i 1) + c(h U, i 1), U V (e) and C(H, x) = (x + 1)C(H e, x) + x C(H U, x). U V (e)

10 Edge decomposition polynomial of hypergraphs Given a hypergraph H(V, E), we call a set S E an edge decomposition of H if every vertex of H belongs to exactly one edge in S. Let d(h, i) denote the number of edge decompositions consisting of i edges of a hypergraph H. Let S H denote the set of all edge decompositions of H. Not every hypergraph has an edge decomposition. Let min S, if S H S S µ(h) = H, otherwise. Then we define the edge decomposition polynomial of H as D(H, x) = E i=µ(h) d(h, i)x i, if S H. 0, otherwise.

11 The recursion restricted to decompositions Theorem 2 Let H(V, E) be a simple hypergraph with rank at least 2 and e be an edge of H, with V (e) denoting the set of vertices of e, then Corollary 3 D(H, x) = D(H e, x) + xd(h V (e), x). (2) If µ(h) =, then µ(h e) = and µ(h V (e)) =, for every edge e of H.

12 Geometric tiling and covering Can think of hypergraph edge covering and decomposition as abstract covering and abstract tiling, respectively. Let R be any collection of finitely many cells from an arbitrary lattice L. We call R a region. Let T be a finite set with each element T consisting of finitely many cells of L. We call each T T a tile. Let T T consist of k cells. Define an admissible position of T as a placement of a congruent copy of T on R such that it fits exactly k cells of R. Problem Given a pair (R, T), count the number of T-tilings and T-coverings of R.

13 Geometric tiling and covering Form a hypergraph H R (V, E) whose vertex set is the set of square cells in R and that has an edge corresponding to every admissible position of T on R, for all T T. Figure 1 : Four admissible positions of a T-tetromino on a 4 4 square grid and the corresponding edges of the hypergraph. Not all admissible positions (edges) are shown.

14 Geometric tiling and covering Theorem 4 The polynomial D(H R, x) is the generating function for the number of T-tilings of R, that is, counts all such tilings. D(H R, 1) = E i=µ(h R ) d(h R, i) Likewise, C(H R, x) is the generating function for the number of T-coverings of R. Corollary 5 Suppose that T = {T }, T consists of k cells and R consists of m cells. Then either k divides m, in which case D(H R, x) = d(h R, m/k)x m/k, or otherwise, D(H R, x) = 0.

15 Ongoing work Existence results Investigate the algebraic and combinatorial properties of D(H, x). Can these properties be used to answer questions about the existence of tilings? Admissibility rules Our present definition of admissible position of a tile is very general for the purpose of tiling. In some cases, it includes positions that cannot be occupied in an actual tiling. More refined admissibility rules should be incorporated. Implementation Implement our combinatorial tiling algorithm and compare its run time with that of the regular grammar algorithm.