Frucht s Algorithm for the Chromatic Polynomial

Transcription

1 Frucht s Algorithm for the Chromatic Polynomial Ramón M. Figueroa-Centeno Universidad Metropolitana Escuela de Matemáticas Apto Caracas. Venezuela. Reinaldo E. Giudici Universidad Simón Bolívar Dpto. de Matemáticas Puras y Aplicadas Apto Caracas. Venezuela. SUMMARY Computing the chromatic polynomial of a graph is an NP-Complete problem. A strategy to count the "Special Spanning Subgraphs", defined by Frucht, is developed as an algorithm. This algorithm is better for some graphs than previous well-known algorithms based in Whitney's Identity found in literature. 1. INTRODUCTION The graphs considered in this paper are always finite, undirected, simple and without loops. The notation and terms not defined here are found in the texts [, ]. A coloring of a graph is an assignment of one specified set {1, 2, 3,... λ} of colors to each point of the graph so that no two adjacent points have the same color. It is well known that the number of different colorings of a graph G in which at most λ colors are used, is a polynomial in λ, called the chromatic polynomial of G, it will be denoted by P(G, λ). Two graphs are said to be chromatically equivalent or χ-equivalent if they have the same chromatic polynomial. It is also well known that computing the chromatic polynomial of a graph, is NP-Hard; furthermore it is NP-Complete [5].Therefore, it is improbable to find an algorithm to compute P(G, λ), that has less than exponential running time. But the chromatic polynomial of a graph turns out to be of theoretical importance because it allows us to characterize a graph through it, hence it is desirable to compute it as fast as possible. The chromatic polynomial of a graph is the product of the chromatic polynomials of its connected components [11], and the multiplication of polynomials is a polynomially bounded problem, thus it is advisable only to consider connected graphs. However this hypothesis will not be required in our considerations, but should be kept in mind by the reader. Definition 1.1 A set S of polynomials is said to span the set of chromatic polynomials C, if any chromatic polynomial can be obtained as a linear combination of the elements of S. A set S that spans C, and whose members are linearly independent is said to be a basis for C. The three most frequently used bases are: (i) The null graph basis: {1, λ, λ 2,..., λ n,...}. (ii) The factorial basis: {1, (λ) 1, (λ) 2,..., (λ) n,...}. (iii) The tree basis: {1, λ(λ - 1), λ (λ - 1) 2,..., λ (λ - 1) n,...}. where (λ) i λ(λ - 1)(λ - 2)... (λ - i + 1), is the falling factorial of i, P(Kn, λ) = (λ) n, and Kn is the complete graph of order n. Hence the chromatic polynomial of a graph G can be written as p P(G, λ) = k i P(K i, λ) (1.1) i = 1 Conversions between these three bases can be obtained using the linear transformations described by Loerinc [6] and Margaglio [7]. Frucht [2] proposed a new way of computing the chromatic polynomial of a graph G, in the factorial basis, by counting the number of certain subgraphs in Ḡ, i.e. the complement of G. Definition 1.2 A Special Spanning Subgraph (SSS) of l components of a graph G is a spanning subgraph of G with l connected components, which are complete graphs.

2 With this definition it is possible to rewrite the chromatic polynomial in terms of the number of SSS s. To aid the reading of this work we state this result as the following theorem. Theorem 1.1 [Frucht, 2] Let G be a graph, then the i-th coefficient of P(G, λ) in the factorial basis, is the number of SSS s of Ḡ with i components. 2. COUNTING THE SPECIAL SPANNING SUBGRAPHS This method has not been popular since, both the author and the reviewer of [2], agree that it takes to much time to count the SSS s. Actually, it took Professor Frucht several hours to compute the chromatic polynomial of the cube graph Q 3, [2, p.75]. Nevertheless, using our technique, the method yields a good algorithm in comparison with the ones previously known for certain types of graphs, i.e. graphs with a large number of lines. First of all, it is remarked that each SSS, of a graph G with p points, and l components obviously corresponds to a partition of the number p in l summands. The converse, is not always true, see [2, p.73]. Thus we shall limit ourselves to write an algorithm for finding all the SSS s with l components, of a graph G with p points, for a given partition P of p in i summands. From now on we associate to a given partition P of a number n in l summands a vector P with l components, in which Pi is the i-th summand of the partition, and Pi Pj, if i j. Our approach is similar to the one found in [], for a clique counting algorithm. Because the number of SSS s of graph can be so large we must be careful in generating them. Each one must be generated just once so that no time is wasted in repetitive work. As in [] we consider first, an unconstrained backtrack search tree for the SSS s, in which each leaf represents finding a SSS of a fixed partition P of n in l summands with associated vector P, or a failure; and each edge represents a point of G. We will build each component of a SSS starting with the largest that is the one that corresponds to P l, in decreasing order until we have a SSS for P. For this purpose let V V(G), this set will help us keep track of which points do we have available at any moment while building a SSS. Let it be the case in which we wish to build the component of the SSS corresponding to Pi = m (1 i l), i.e. a Km, with the points in V. Let N1 V. Take a point a N1 and let N1 (N1 Adj(a)), where Adj(a) is the set of points adjacent to a V, and V (V - {a}). Repeat the last step at most m times or until N1 = Ø. If N1 = Ø before m steps, then the corresponding branch of the tree does not produce a Km, and thus no SSS is obtained. If we arrive to the end of the m steps, we would have built a Km, whose points are the m points we just deleted from V. If we succeed doing the above process for each Pi, we would have that V = Ø and a SSS is built. Note that if Pi = 1 then the process always succeeds, thus it suffices only to consider Pi 1. In order to prune the backtrack search tree, we use a modified version of Theorem.2 of []. Theorem 2.1 Let x V, be a point chosen to be a root of a Kn, then all the Kn s with points in V containing x, will be generated in the next (n-1) steps of the recursion. Thus we need not consider {x} to build the following Kn s. The above theorem applies for any K Pi, for any i such that P i = n, if we fix i we have the following result. Corollary 2.1 Let it be the case in which we want to build a K P i in particular, if we choose x V as a root of K P i. Then all the K P i s with points in V containing x, will be generated in the next (n-1) steps of the recursion. Thus we need only consider v Adj(x) V to construct the other possible K Pi. To keep track of which points have been used as roots of Kn s, we use a set N2. And the algorithm is obtained.

3 An example of a graph G and part of it s backtrack search tree T, induced by the partition P = (1, 2, 2, 3) is shown in Figures 1-2. Note that each set, in the leafs of T, is generated several times. All the edges of T shown in lightface can be pruned using Theorem 2.1 and Corollary 2.1. The number of SSS s of G induced by P, is the number of leaves of T in the last level, after the pruning FIGURE 1

4 ø {1} {2} {3} {} {5} {6} {7} {} K {1,2}{1,3}{1,5}{1,2}{2,3}{2,7}{1,3}{2,3}{3,}{3,}{,5}{1,5}{,5}{5,6}{5,} {5,6}{6,7} {2,7}{6,7}{5,} {1,2,3}{1,2,3} {1,2,3}{1,2,3} {1,2,3}{1,2,3} * * K2 {} {5} {6} {7} {} {,5} {,5} {5,6} {5,} {5,6} {6,7} {6,7} {5,} K2 {6} {7} {} {} {7} {} {} {6} {7} {} {5} {} {6,7}{6,7} {6,7}{6,7} {,5}{,5}{5,}{5,} K1 {} {} {} {} {} {} {} {} FIGURE 2

5 ALGORITHM 2.1: Input: G : undirected simple graph. max: maximum clique of G. l: order of of the SSS of G we are looking for. P: a partition of V(G) in l components. first = min{i : Pi 1} Output: S: number of SSS s of G of order l induced by P. S 0 V V(G) if l = V then S 1 else if Pl max then SSS(V, V, size, 1) Output(S) procedure SSS (N1, N2 : set of points; i, depth : integer) begin if (depth = Pi) and (i = first) then S S + N1 else while N1 Ø do begin u point in N1 N1 N1 - {u} if depth = 1 then N2 N2 - {u} V V - {u} if (depth = Pi) then begin if (Pi Pi-1) then SSS(V, V, i - 1, 1) else SSS(N2 V, N2, i - 1, 1) end else SSS(N1 Adj(u), N2, i, depth + 1) V V {u} end end It is easy to prove that Algorithm 2.1 is correct. 3.- SOME CONCLUSIONS We have not done the running time analysis of the algorithm but certain considerations shed some light upon it. We state the following two obvious results, which indeed are consequences of Theorem 1.1. Theorem 3.1 Let G and H be graphs whose chromatic polynomials have as i-th coefficient in the factorial basis gi and hi respectively, and such that V(H) = V(G), and E(H) E(G). Then hi gi, for each i. Corollary 3.1 The graph with p points with the largest coefficients in the factorial basis is the null graph Np.

6 It is a well known fact that the i-th coefficient of P(Np, λ) in the factorial basis is S(p,i), a Stirling number of the second kind [9]. In particular, S(p,2) = 2 p-1-1, which means that the number of SSS s of Kp is exponential. And thus the running time of the algorithm can be exponential. Another more interesting question, which we leave open, is if the algorithm is exponential on it s output, such as the clique counting algorithm found in []? An implementation of the algorithm has been compared against the best implementation known to the authors of a Whitney s Identity based algorithm, see [1].This implementation runs faster for some graphs with large number of points, e.g. K 7 7 and K. We must note that the data structures used are relatively simple compared to the ones used in Whitney s Identity based algorithms, in which there is a generation of an exponential number of graphs in the recursive process. Thus the computational space used by Algorithm 2.1 is less than the one used in those algorithms. We also have that the operations done on the structures are relatively simple, such as basic set operations, compared to ones such as DFS searches and identification of points, see [1]. The following question arises, when is Frucht s method useful? Sometimes the case comes forth in which we only need to know the first coefficients of P(G, λ) in the factorial basis, e.g. finding if two graphs are not χ-equivalent or the chromatic number χ(g) of a graph G. Frucht s method allows us to compute the chromatic polynomial in the factorial basis one coefficient at a time, this makes it particularly appropriate when only a few of the first, or last, coefficients suffice. The Algorithm 2.1 was implemented using Turbo Pascal 1.1, on a Apple Macintosh computer. On a Macintosh SE/30 with 5Mb of RAM memory, it takes roughly 19 seconds to compute the chromatic polynomial of the Heawood graph. Using the notation given in [3], the coefficients obtained are the following. N: T: F:

7 For the graph Q 3 the algorithm takes 0.7 seconds N: T: F: ACKNOLEDGEMENT The authors gratefully wish to thank Professor Claudio Margaglio for providing the highly efficient code used, for the chromatic polynomial basis transformations. BIBLIOGRAPHY [1] Figueroa-Centeno, R.M. and Giudici, R.E., An Improved Algorithm for The Chromatic Polynomial, Reporte No. 1-9, Escuela de Matemáticas, Universidad Metropolitana (199). [2] Frucht, R. W., A New Method Of Computing Chromatic Polynomials of Graphs, Analysis, Geometry, and Probability,Chuaqui, R., ed.(marcel Dekker, Inc., New York, New York, 195, 69-77). [3] Giudici, R.E. and Vinke, R.M., A Table of Chromatic Polynomials, J. of Comb. Infor. and Syst Sc., 5 (190) [] Harary, F., Graph Theory (Addison-Wesley, Reading, MA, 1969). [5] Karp, R.M., Reductibility among combinatorial problems, in Complexity of Computer Computations, Miller, R.E. and Thatcher, J.M., eds. (Plenum Press, 1972) [6] Loerinc, B.M., Computing chromatic polynomials for special families of graphs, Courant Computer Science Report No. 19 (Courant Inst. of Math. Sciences, Computer Science Dept., New York Univ. February 190). [7] Margaglio, C., Un Programa para el Cambio de Base de Polinomios Cromáticos, Reporte No. 9-0 Dpto. de Matemáticas Puras y Aplicadas Universidad Simón Bolívar (199). [] Reingold, E.M., Nievergelt, J. and Deo, N., Combinatorial Algorithms (Prentice-Hall, Englewoods Cliffs, NJ, 1977). [9] Whitehead, E.G., Jr., Stirling number identities from chromatic polynomials, J. of Combinatorial Theory, Series A, 2 (197), ). [10] Whitney, H., The coloring of graphs, Ann. Math.,33(1932) [11] Zykov,A.A., On some properties of linear complexes, Amer. Math. Soc. Transl. Nº79 (1952); translated from Math. Sb., 2, Nº66 (199),163-1.