Crossing Numbers of Random Graphs
Joel Spencer, Géza Toth
a lecture by Puck Rombach
The crossing number, denoted by cr(G), of G is the minimum number of crossing points in any drawing of G.
The rectilinear crossing number, denoted by lin-cr(G), of G is the minimum number of crossing points in any drawing of G, with straight line segments as edges.
The pairwise crossing number, denoted by pair-cr(G), of G is the minimum number
of pairs of crossing edges over all drawings of G.
Let
γcr = lim
n→∞
cr(Kn )
n 2
2
and define γlin-cr and γpair-cr in a similar way. Let
κcr (n, p) =
E(cr(G))
e2
where e = p n2 is the expected number of edges of G, and define κlin-cr (n, p) and
κpair-cr (n, p) in a similar way.
An order type of the points x1 , x2 , ..., xn in the plane (with no three colinear) is a list
of orientations of all triplet xi xj xk , i < j < k. Let X be the set of all order types of
the points x1 , ..., xn in the plane. For any graph G with vertices v1 , ..., vn let lin-crξ (G)
denote the number of crossings in the straight line drawing of G, where vi is placed at xi
in the plane and x1 , ..., xn have order type ξ ∈ X.
Theorem 1.
For any fixed n, κlin-cr (n, p), κcr (n, p) and κpair-cr (n, p) are increasing, continuous functions of p.
Theorem 2.
1. lim supn→∞ κlin-cr (n, c/n) = 0
f or
2. lim supn→∞ κcr (n, c/n) = 0 f or
c≤1
c≤1
3. lim supn→∞ κpair-cr (n, c/n) = 0 f or c ≤ 1
4. limc→1 lim supn→∞ κlin-cr (n, c/n) = 0
5. limc→1 lim supn→∞ κcr (n, c/n) = 0
1
6. limc→1 lim supn→∞ κpair-cr (n, c/n) = 0
7. lim supn→∞ κlin-cr (n, c/n) < γlin-cr f or all c
8. lim supn→∞ κcr (n, c/n) < γcr f or all c
9. lim supn→∞ κpair-cr (n, c/n) < γpair-cr f or all c.
Theorem 3.
For any ε > 0, p = p(n) = nε−1 ,
lim inf κpair-cr (n, p) > 0.
n→∞
Theorem 4.
For any c > 1 with p = p(n) = c/n
lim inf κcr (n, p) > 0.
n→∞
Theorem 5.
If p = p(n) log n
,
n
then
lim κlin-cr (n, p) = γlin-cr (n, p).
n→∞
Define the bisection width of G, denoted by b(G), as the minimal number of edges running
between T (top) and B (bottom) over all partitions of the vertex set into two disjoint
parts V = T ∪ B such that 32 |V | ≥ |T |, |B| ≥ 31 |V |.
Theorem 6.
Let V be a set of m vertices. Let T be a tree on V . Let G be the random graph on
a
V with edge probability p = m
. For a > 0 fixed almost surely
b(T ∪ G) = Ω(m).
That is, there exists η > 0 dependent only on a such that P(b(T ∪ G) ≤ mη) approaches
zero as m approaches infinity.
An order type of the points x1 , x2 , ..., xn in the plane (with no three colinear) is a list
of orientations of all triplet xi xj xk , i < j < k. Let X be the set of all order types of
the points x1 , ..., xn in the plane. For any graph G with vertices v1 , ..., vn let lin-crξ (G)
denote the number of crossings in the straight line drawing of G, where vi is placed at xi
in the plane and x1 , ..., xn have order type ξ ∈ X.
Theorem 7.
Let G(n, p) be a random graph with vertices v1 , v2 , ..., vn , with edge probability 0 < p =
p(n) < 1. Then
P |lin-crξ (G) − E(lin-crξ (G))| > 3αe3/2 < 3 exp(−α2 /4)
√
holds for every α satisfying (e/4)3 exp(−e/4) ≤ α ≤ e.
2