Mathematica Notebook accompanying the article M. Gerling: Bivariate Chromatic Polynomials in Computer Algebra
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1 Mathematica Notebook accompanying the article M. Gerling: Bivariate Chromatic Polynomials in Computer Algebra SetDirectory["..."] << BivariatePolynomials.m To compute the first example for Theorem 11, we first define the given graph. ShowGraph[G1 = DeleteEdges[Combinatorica`CompleteGraph[9], {{1, 2}, {1, 3}, {1, 4}, {5, 6}, {5, 7}, {8, 9}}]] To compute the bivariate chromatic polynomial for this graph, we use the implementation "BivariatePolynomialAGM1" of the recursion formula (AGM1) for a graph G by Averbouch, Godlin and Makowsky from Theorem 6. As we see, this method needs some minutes on a PC. Timing[BivariatePolynomialAGM1[G1, x, y]] Dauer , x 9-30 x 7 y x 6 y x 5 y x 5 y x 4 y x 4 y x 3 y x 3 y x 3 y x 2 y x 2 y x 2 y xy xy xy xy-1114 y y y y Now, we use the implementation "BivariatePolynomialAGM2" for the same
2 recursion formula (AGM2), which solves the equation for the graph G - e instead of G with an edge e from E. For our graph, this is much faster than the implementation "BivariatePolynomialAGM1". Timing[BivariatePolynomialAGM2[G1, x, y]] Dauer , x 9-30 x 7 y x 6 y x 5 y x 5 y x 4 y x 4 y x 3 y x 3 y x 3 y x 2 y x 2 y x 2 y xy xy xy xy-1114 y y y y Our equation from Theorem 11 is implemented directly and is clearly more efficient than the implementation "BivariatePolynomialAGM1". Timing[ Expand[ Sum[ Binomial[1, c]*3^c* Sum[ Binomial[1, b]*2^b* Dauer multipliz s Binomialkoeffizient s Binomialkoeffizient Sum[ Binomial[1, a]*1^a* Sum[ Binomial[9-2 * c - 2 * b - 2 * a, i]* s Binomialkoeffizient s Binomialkoeffizient (x - y)^i* FunctionExpand[ FactorialPower[y, 9 - c - b - a - i]], multipliziere Funktio Faktorielle {i, 0, 9-2 * c - 2 * b - 2 * a}], {a, 0, 1}], {b, 0, 1}], {c, 0, 1}]]] , x 9-30 x 7 y x 6 y x 5 y x 5 y x 4 y x 4 y x 3 y x 3 y x 3 y x 2 y x 2 y x 2 y xy xy xy xy-1114 y y y y Now we consider the second example for Theorem 11. Because of the graph' s size we define the graph without showing it. G2 = DeleteEdges[Combinatorica`CompleteGraph[40], {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {7, 8}, {7, 9}, {7, 10}, {7, 11}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {17, 18}, {17, 19}, {17, 20}, {21, 22}, {23, 24}, {25, 26}, {27, 28}}]; In this case we use only the implementation "BivariatePolynomialAGM2", because it is more efficient for the graph G2. But even this recursion cannot compute the bivariate chromatic polynomial in less than one hour. TimeConstrained[BivariatePolynomialAGM2[G2, x, y], 3600] zeitbeschränkt $Aborted
3 Our equation from Theorem 11 can solve this problem easily. However, the output is very large. Timing[ Expand[ Dauer multipliziere aus Sum[ Binomial[1, e]*5^e* s Binomialkoeffizient Sum[ summiere Binomial[2, d]*4^d* Sum[ Binomial[1, c]*3^c* Sum[ Binomial[0, b]*2^b* Sum[ Binomialkoeffizient s Binomialkoeffizient s Binomialkoeffizient summiere Binomial[4, a]*1^a* Sum[ Binomial[40-2 * e - 2 * d - 2 * c - 2 * b - 2 * a, i]* Binomialkoeffizient s Binomialkoeffizient (x - y)^i* FunctionExpand[ FactorialPower[y, 40 - e - d - c - b - a - i]], multipliziere Funktio Faktorielle {i, 0, 40-2 * e - 2 * d - 2 * c - 2 * b - 2 * a}], {a, 0, 4}], {b, 0, 0}], {c, 0, 1}], {d, 0, 2}], {e, 0, 1}]]] , x yx yx y 2 x yx y 2 x yx y 3 x y 2 x yx y 3 x y 2 x yx y 4 x y 3 x y 2 x yx y 4 x y 3 x y 2 x yx y 5 x y 4 x y 3 x y 2 x yx y 5 x y 4 x y 3 x y 2 x yx y 6 x y 5 x y 4 x y 3 x y 2 x yx y 6 x y 5 x y 4 x y 3 x y 2 x yx y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 9 x y 8 x 21 +
4 y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 13 x y 12 x y 11 x y 10 x 13 +
5 y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x 9 -
6 y 6 x y 5 x y 4 x y 3 x y 2 x yx y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx 6 +
7 y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 18 x y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 18 x y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 19 x 2 +
8 y 18 x y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 19 x y 18 x y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y y y y y y y y y y y y y y y y y y 3 +
9 y y We turn to the first example for Theorem 13. ShowGraph[G3 = CompleteKPartiteGraph[2, 2, 2, 2]] As above, we first compute the bivariate chromatic polynomial by the implementation "BivariatePolynomialAGM1". Timing[BivariatePolynomialAGM1[G3, x, y]] Dauer , x 8-24 x 6 y + 88 x 5 y x 4 y x 4 y x 3 y x 3 y x 2 y x 2 y x 2 y xy xy xy+ 60 y y y y The implemenation "BivariatePolynomialAGM2" needs less time to compute the same polynomial. Timing[BivariatePolynomialAGM2[G3, x, y]] Dauer , x 8-24 x 6 y + 88 x 5 y x 4 y x 4 y x 3 y x 3 y x 2 y x 2 y x 2 y xy xy xy+ 60 y y y y Our own equation from Theorem 13 needs less time than the implementation "BivariatePolynomialAGM1". Timing Expand Sum Binomial[4, i]* Sum Binomial[8-2 * i, l]*(x-y)^ 8-2 * i - l * Dauer multipliz s Binomialkoeffizient s Binomialkoeffizient FunctionExpand[ FactorialPower[y, i + l]], {l, 0, 8-2 * i}, {i, 0, 4} multipliziere Funktio Faktorielle 0., x 8-24 x 6 y + 88 x 5 y x 4 y x 4 y x 3 y x 3 y x 2 y x 2 y x 2 y xy xy xy+ 60 y y y y
10 The second example for Theorem 13 is a large graph, so we don' t depict it. ShowGraph[G4 = CompleteKPartiteGraph[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]]; We see that not even the implementation "BivariatePolynomialAGM2", which is much more advantageous for this graph, can compute the bivariate chromatic polynomial in less than one hour. TimeConstrained[BivariatePolynomialAGM2[G4, x, y];, 3600] zeitbeschränkt $Aborted The formula from Theorem 13 can do the computation in few seconds. However, the output is very large. Timing Dauer Expand Sum Binomial[30, i]* Sum Binomial[60-2 * i, l]*(x-y)^ 60-2 * i - l * multipliz s Binomialkoeffizient s Binomialkoeffizient FunctionExpand[ FactorialPower[y, i + l]], {l, 0, 60-2 * i}, {i, 0, 30} multipliziere Funktio Faktorielle , x yx yx y 2 x yx y 2 x yx y 3 x y 2 x yx y 3 x y 2 x yx y 4 x y 3 x y 2 x yx y 4 x y 3 x y 2 x yx y 5 x y 4 x y 3 x y 2 x yx y 5 x y 4 x y 3 x y 2 x yx y 6 x y 5 x y 4 x y 3 x y 2 x yx y 6 x y 5 x y 4 x y 3 x y 2 x yx y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x 44 -
11 yx y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x 36 +
12 y 2 x yx y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x 32 +
13 y 2 x yx y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 16 x 28 -
14 y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x 26 +
15 y 2 x yx y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 18 x y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 18 x y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x 23 +
16 y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 19 x y 18 x y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x yx y 19 x y 18 x y 17 x y 16 x y 15 x y 14 x y 13 x y 12 x y 11 x y 10 x y 9 x y 8 x y 7 x y 6 x y 5 x y 4 x y 3 x y 2 x 21 +
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