e11-property-relation-conjectures.sagews

#generating lower boind conjectures for being squarefree
#(sufficent comdition conjectures for being squarefree)
load("conjecturing.py")

def is_perfect(n):
    return sigma(n) == 2*n

objects = [2,6,8,47,1000]
properties = [is_prime,is_prime_power,is_perfect,is_squarefree,is_triangular_number,is_odd,is_even]
propertyBasedConjecture(objects,properties,properties.index(is_squarefree))

[(is_prime)->(is_squarefree), (is_perfect)->(is_squarefree)]

#28 is a CE to the 2nd conjecture, 28 to our objects
load("conjecturing.py")

def is_perfect(n):
    return sigma(n) == 2*n

objects = [2,6,8,47,1000,28]
properties = [is_prime,is_prime_power,is_perfect,is_squarefree,is_triangular_number,is_odd,is_even]
propertyBasedConjecture(objects,properties,properties.index(is_squarefree))

[(is_prime)->(is_squarefree)]

#want necessary condition conjectures, add parament "suffient = False"
load("conjecturing.py")

def is_perfect(n):
    return sigma(n) == 2*n

objects = [2,6,8,47,1000]
properties = [is_prime,is_prime_power,is_perfect,is_squarefree,is_triangular_number,is_odd,is_even]
propertyBasedConjecture(objects,properties,properties.index(is_squarefree), sufficient = False)

[(is_squarefree)->((is_perfect)|(is_prime))]

pete = graphs.PetersenGraph()

pete.show()

pete.is_hamiltonian()

False

load("conjecturing.py")

k3 = graphs.CompleteGraph(3)
k4 = graphs.CompleteGraph(4)
k5 = graphs.CompleteGraph(5)
p3 = graphs.PathGraph(3)

def max_degree(g):
    return max(g.degree())

def min_degree(g):
    return min(g.degree())

def is_dirac(g):
    n = g.order()
    return min_degree(g) >= n/2

objects = [k3, k4, k5, p3]
properties = [Graph.is_hamiltonian, max_degree, min_degree, is_dirac, Graph.is_tree, Graph.is_planar, Graph.is_connected]
propertyBasedConjecture(objects, properties, properties.index(Graph.is_hamiltonian))

[(is_dirac)->(is_hamiltonian)]

load("conjecturing.py")

k3 = graphs.CompleteGraph(3)
k4 = graphs.CompleteGraph(4)
k5 = graphs.CompleteGraph(5)
p3 = graphs.PathGraph(3)

def max_degree(g):
    return max(g.degree())

def min_degree(g):
    return min(g.degree())

def is_dirac(g):
    n = g.order()
    return min_degree(g) >= n/2

objects = [k3, k4, k5, p3]
properties = [Graph.is_hamiltonian, max_degree, min_degree, is_dirac, Graph.is_tree, Graph.is_planar, Graph.is_connected]
propertyBasedConjecture(objects, properties, properties.index(Graph.is_hamiltonian), sufficient = False)

[(is_hamiltonian)->(is_dirac)]

c5 = graphs.CycleGraph(5)
c5.show()

#C5, the cycle on 5 vertics, is a CE to the last conjecture
#C5 is hamiltonian, min_degree = 2, order = 5 (not dirac!)

load("conjecturing.py")

k3 = graphs.CompleteGraph(3)
k4 = graphs.CompleteGraph(4)
k5 = graphs.CompleteGraph(5)
p3 = graphs.PathGraph(3)
c5 = graphs.CycleGraph(5)

def max_degree(g):
    return max(g.degree())

def min_degree(g):
    return min(g.degree())

def is_dirac(g):
    n = g.order()
    return min_degree(g) >= n/2

objects = [k3, k4, k5, p3, c5]
properties = [Graph.is_hamiltonian, is_dirac, Graph.is_tree, Graph.is_planar, Graph.is_connected]
propertyBasedConjecture(objects, properties, properties.index(Graph.is_hamiltonian), sufficient = False)

[(is_hamiltonian)->(~(is_tree))]

L= [3,4,5]
mean(L)

4

mean(c5.degree())

2