***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001725 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- Description: The harmonious chromatic number of a graph. A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Harmonious_coloring]] ----------------------------------------------------------------------------- Code: def statistic(G): G = G.canonical_label().copy(immutable=True) return statistic_aux(G) @cached_function def statistic_aux(G): """ sage: N = 8; lG = [G for n in range(1, N) for G in graphs(n)] sage: all(statistic(G) >= max(G.degree()) + 1 for G in lG) True sage: all(statistic(G) <= 2*max(G.degree())*sqrt(G.num_verts()-1) for G in lG if G.edges()) True sage: all((statistic(G) == G.num_verts()) == (G.diameter() <= 2) for G in lG) True sage: all(statistic(graphs.CompleteGraph(n)) == n for n in range(6)) True sage: def min_k(G): ....: k = 0 ....: while True: ....: if binomial(k, 2) >= G.num_edges(): ....: return k ....: k += 1 sage: def harmonious_path(n): ....: k = min_k(graphs.PathGraph(n)) ....: if is_odd(k) or (k-2)//2 <= binomial(k, 2) - n + 1 <= k-2: ....: return k ....: return k+1 sage: all(statistic(graphs.PathGraph(n)) == harmonious_path(n) for n in range(1, 8)) True """ G = G.relabel(inplace=False) n = G.num_verts() K = range(n) # colors Kp = [(i, j) for i in K for j in range(i+1, n)] # pairs of colours V = G.vertices() E = [tuple(sorted(e)) for e in G.edges(labels=False)] P = MixedIntegerLinearProgram(maximization=False) # y[c] == 1 if c is used y = P.new_variable(binary=True, indices=K) # x[(v,c)] == 1 if v is colored with c x = P.new_variable(binary=True, indices=cartesian_product([V, K])) # z[e, c, d] == 1 if the edge e is incident to colours c < d z = P.new_variable(binary=True, indices=cartesian_product([E, Kp])) P.set_objective(sum(y[c] for c in K)) for v in V: # one color per node P.add_constraint(sum(x[(v,c)] for c in K) == 1) for c in K: # if vertex v takes color c, activate y[c] P.add_constraint(x[(v,c)] <= y[c]) for u, v in E: for c in K: # the colouring is proper P.add_constraint(x[(u,c)] + x[(v,c)] <= 1) if E: for c, d in Kp: P.add_constraint(x[(u,c)] + x[(v,d)] <= 1 + z[((u, v), (c, d))]) P.add_constraint(x[(u,d)] + x[(v,c)] <= 1 + z[((u, v), (c, d))]) if E: for c, d in Kp: P.add_constraint(sum(z[((u, v), (c, d))] for u, v in E) <= 1) return ZZ(P.solve()) ----------------------------------------------------------------------------- Statistic values: ([],1) => 1 ([],2) => 1 ([(0,1)],2) => 2 ([],3) => 1 ([(1,2)],3) => 2 ([(0,2),(1,2)],3) => 3 ([(0,1),(0,2),(1,2)],3) => 3 ([],4) => 1 ([(2,3)],4) => 2 ([(1,3),(2,3)],4) => 3 ([(0,3),(1,3),(2,3)],4) => 4 ([(0,3),(1,2)],4) => 3 ([(0,3),(1,2),(2,3)],4) => 3 ([(1,2),(1,3),(2,3)],4) => 3 ([(0,3),(1,2),(1,3),(2,3)],4) => 4 ([(0,2),(0,3),(1,2),(1,3)],4) => 4 ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4 ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4 ([],5) => 1 ([(3,4)],5) => 2 ([(2,4),(3,4)],5) => 3 ([(1,4),(2,4),(3,4)],5) => 4 ([(0,4),(1,4),(2,4),(3,4)],5) => 5 ([(1,4),(2,3)],5) => 3 ([(1,4),(2,3),(3,4)],5) => 3 ([(0,1),(2,4),(3,4)],5) => 3 ([(2,3),(2,4),(3,4)],5) => 3 ([(0,4),(1,4),(2,3),(3,4)],5) => 4 ([(1,4),(2,3),(2,4),(3,4)],5) => 4 ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 5 ([(1,3),(1,4),(2,3),(2,4)],5) => 4 ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 4 ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4 ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 4 ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 5 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5 ([(0,4),(1,3),(2,3),(2,4)],5) => 4 ([(0,1),(2,3),(2,4),(3,4)],5) => 4 ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 4 ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 5 ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 5 ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 5 ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 5 ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 4 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4 ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 5 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 5 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5 ([],6) => 1 ([(4,5)],6) => 2 ([(3,5),(4,5)],6) => 3 ([(2,5),(3,5),(4,5)],6) => 4 ([(1,5),(2,5),(3,5),(4,5)],6) => 5 ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 6 ([(2,5),(3,4)],6) => 3 ([(2,5),(3,4),(4,5)],6) => 3 ([(1,2),(3,5),(4,5)],6) => 3 ([(3,4),(3,5),(4,5)],6) => 3 ([(1,5),(2,5),(3,4),(4,5)],6) => 4 ([(0,1),(2,5),(3,5),(4,5)],6) => 4 ([(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 5 ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(2,4),(2,5),(3,4),(3,5)],6) => 4 ([(0,5),(1,5),(2,4),(3,4)],6) => 4 ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4 ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 4 ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 5 ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,4),(2,3)],6) => 3 ([(1,5),(2,4),(3,4),(3,5)],6) => 4 ([(0,1),(2,5),(3,4),(4,5)],6) => 4 ([(1,2),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 4 ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 4 ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 5 ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 6 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 5 ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 4 ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 5 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 4 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 4 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 5 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 5 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 4 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 5 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 5 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 6 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 5 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 5 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 5 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 5 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 6 ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 5 ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 5 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 6 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 6 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 6 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 6 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 6 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([],7) => 1 ([(5,6)],7) => 2 ([(4,6),(5,6)],7) => 3 ([(3,6),(4,6),(5,6)],7) => 4 ([(2,6),(3,6),(4,6),(5,6)],7) => 5 ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 7 ([(3,6),(4,5)],7) => 3 ([(3,6),(4,5),(5,6)],7) => 3 ([(2,3),(4,6),(5,6)],7) => 3 ([(4,5),(4,6),(5,6)],7) => 3 ([(2,6),(3,6),(4,5),(5,6)],7) => 4 ([(1,2),(3,6),(4,6),(5,6)],7) => 4 ([(3,6),(4,5),(4,6),(5,6)],7) => 4 ([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 5 ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 5 ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 6 ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(3,5),(3,6),(4,5),(4,6)],7) => 4 ([(1,6),(2,6),(3,5),(4,5)],7) => 4 ([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4 ([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 4 ([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => 4 ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4 ([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 4 ([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 4 ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 5 ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 5 ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 5 ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => 5 ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 5 ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 5 ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => 5 ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 7 ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(1,6),(2,5),(3,4)],7) => 3 ([(2,6),(3,5),(4,5),(4,6)],7) => 4 ([(1,2),(3,6),(4,5),(5,6)],7) => 4 ([(0,3),(1,2),(4,6),(5,6)],7) => 4 ([(2,3),(4,5),(4,6),(5,6)],7) => 4 ([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 4 ([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => 4 ([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 4 ([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => 4 ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 5 ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 5 ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 7 ([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => 5 ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4 ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4 ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 5 ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7) => 5 ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4 ([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7) => 4 ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 5 ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7) => 5 ([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7) => 5 ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(1,6),(2,5),(3,4),(3,5),(4,6)],7) => 4 ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 4 ([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => 4 ([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 4 ([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => 5 ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4 ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4 ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7) => 5 ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => 4 ([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 4 ([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4 ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 4 ([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 4 ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 4 ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => 5 ([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 5 ([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7) => 5 ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 5 ([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 5 ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 6 ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 6 ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 4 ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4 ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => 5 ([(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 5 ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 6 ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7) => 5 ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 5 ([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7) => 6 ([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7) => 5 ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 5 ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7) => 6 ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7) => 5 ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 5 ([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7) => 5 ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7) => 6 ([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7) => 5 ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => 5 ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 6 ([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7) => 5 ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 5 ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 5 ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5 ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 6 ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7 ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7) => 6 ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 7 ----------------------------------------------------------------------------- Created: Jun 03, 2021 at 09:53 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jun 03, 2021 at 09:53 by Martin Rubey