This method is based on work of Francisco-Ha-Van Tuyl, looking at the associated primes of the square of the Alexander dual of the edge ideal. The algorithm counts the number of these associated primes of height 3.
i1 : R = QQ[x_1..x_6]; |
i2 : G = graph({x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_1*x_5,x_1*x_6,x_5*x_6}) --5-cycle and a triangle o2 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}} 1 2 2 3 3 4 1 5 4 5 1 6 5 6 ring => R vertices => {x , x , x , x , x , x } 1 2 3 4 5 6 o2 : Graph |
i3 : numTriangles G o3 = 1 |
i4 : H = completeGraph R; |
i5 : numTriangles H == binomial(6,3) o5 = true |