An odd hole is an odd induced cycle of length at least 5. The 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.
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 : hasOddHole G o3 = true |
i4 : H = 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,x_1*x_4}) --no odd holes
o4 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
1 2 2 3 1 4 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
o4 : Graph
|
i5 : hasOddHole H o5 = false |