令d表示图中顶点的最小程度。对于d≥3的n个顶点上的所有平面图,下列哪一项是正确的?
(A)在任何平面嵌入中,面的数量至少为n / 2 + 2
(B)在任何平面嵌入中,面的数量小于n / 2 + 2
(C)有一个平面嵌入,其中的面数小于n / 2 + 2
(D)存在平面嵌入,其中面数最多为n /(d + 1)答案: (A)
解释:
Euler's formula for planar graphs:
v − e + f = 2.
v → Number of vertices
e → Number of edges
f → Number of faces
Since degree of every vertex is at least 3,
below is true from handshaking lemma (Sum of
degrees is twice the number of edges)
3v ≤ 2e
3v/2 ≤ e
Putting these values in Euler's formula.
v - 3v/2 + f ≥ 2
f ≥ v/2 + 2
这个问题的测验