The crossing number of the cone of a graph C. A. Alfaro, A. Arroyo, - - PowerPoint PPT Presentation

The crossing number of the cone of a graph

• C. A. Alfaro, A. Arroyo, M. Derˇ

n´ ar, B. Mohar GD2016, Athens, Greece, September 20

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Cone of a graph

Cone of G: C(G) . . . add a universal vertex (apex)

apex G C(G)

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Motivation: Two conjectures and the cone

◮ Harary-Hill Conjecture:

cr(Kn) =

• 1

64n(n − 2)2(n − 4),

n is even;

1 64(n − 1)2(n − 3)2,

n is odd.

◮ Albertson’s Conjecture: χ(G) ≥ r

⇒ cr(G) ≥ cr(Kr)

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Motivation: Two conjectures and the cone

◮ Harary-Hill Conjecture:

cr(Kn) =

• 1

64n(n − 2)2(n − 4),

n is even;

1 64(n − 1)2(n − 3)2,

n is odd.

◮ Albertson’s Conjecture: χ(G) ≥ r

⇒ cr(G) ≥ cr(Kr)

1 In the H-H Conjecture: Kn+1 = C(Kn). 2 In a special case for Albertson’s conjecture, χ(C(G)) = χ(G) + 1.

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Motivation: Two conjectures and their variation

Albertson’s Conjecture: χ(G) ≥ r ⇒ cr(G) ≥ cr(Kr)

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Motivation: Two conjectures and their variation

Albertson’s Conjecture: χ(G) ≥ r ⇒ cr(G) ≥ cr(Kr) A variation asked by Bruce Richter: Given n ≥ 5 and a graph G with cr(G) ≥ cr(Kn), does it follow that cr(CG) ≥ cr(Kn+1)?

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Richter’s question

cr(G) ≥ cr(Kn) ⇒ cr(CG) ≥ cr(Kn+1)? Observation: True for n = 5. (Because C(K5) = K6 and cr(C(K3,3)) = 3)

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Richter’s question

cr(G) ≥ cr(Kn) ⇒ cr(CG) ≥ cr(Kn+1)? Observation: True for n = 5. (Because C(K5) = K6 and cr(C(K3,3)) = 3) False for n = 6. cr(G) = 3 = cr(K6) and cr(C(G)) = 6 < cr(K7) = 9

Alfaro-Arroyo-Dernar-M. cr(cone(G))

New question

Problem: Suppose cr(G) = k. How much bigger is cr(C(G))? φ(k) := min{cr(C(G)) − cr(G) | cr(G) = k}

Alfaro-Arroyo-Dernar-M. cr(cone(G))

New question

Problem: Suppose cr(G) = k. How much bigger is cr(C(G))? φ(k) := min{cr(C(G)) − cr(G) | cr(G) = k} Theorem.

• k/2 ≤ φ(k) ≤

√ 3k Upper bound: Previous graph with edge-multiplicities r has crossing number 3r2 and its cone has crossing number 3r2 + 3r.

Alfaro-Arroyo-Dernar-M. cr(cone(G))

The lower bound

• Theorem. cr(C(G)) ≥ cr(G) +
• 1

2cr(G)

• Proof. Step 1: Obtain a 1-page drawing of G:

a

• D

e1 e2 e3 e4 e5 e6 D D0

(a) (b) (c)

Alfaro-Arroyo-Dernar-M. cr(cone(G))

The lower bound

• Theorem. cr(C(G)) ≥ cr(G) +
• 1

2cr(G)

Proof (cont’d). From a 1-page drawing to a 2-page drawing.

Lemma (Edwards 1975)

G graph of order n with m ≥ 1 edges. Then G contains a bipartite subgraph with at least 1

2m +

• 1

8m + 1 64 − 1 8 > 1 2m edges.

Corollary

Let D be a 1-page drawing of a graph G with k ≥ 1 crossings. Then some edges of G can be redrawn in a new page, obtaining a 2-page drawing with ≤ 1

2k −

• 1

8k + 1 64 + 1 8 crossings.

Such a drawing can be found in time O(|E(G)| + k) (Bollob´ as & Scott, 2002).

Alfaro-Arroyo-Dernar-M. cr(cone(G))

The lower bound

• Theorem. cr(C(G)) ≥ cr(G) +
• 1

2cr(G)

Proof (cont’d).

D0

If too few crossings, we obtain a drawing of G with < cr(G) crossings.

• Alfaro-Arroyo-Dernar-M.

cr(cone(G))

Simple graphs

φs(k) := min{cr(C(G)) − cr(G) | cr(G) = k, G is simple} The lower bound φs(k) ≥

• k/2 still holds.
• Theorem. φs(3) = 3, φs(4) = 4, φs(5) = 5.

Obvious conjecture!

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Simple graphs

φs(k) := min{cr(C(G)) − cr(G) | cr(G) = k, G is simple} The lower bound φs(k) ≥

• k/2 still holds.
• Theorem. φs(3) = 3, φs(4) = 4, φs(5) = 5.

Obvious conjecture!

• Theorem. φs(k) = O(k3/4)

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Conjecture

Conjecture φs(k) = √ 2 k3/4(1 + o(1))

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Conjecture

Conjecture φs(k) = √ 2 k3/4(1 + o(1)) This specific form of the conjecture is due to the following

• bservation:

Proposition

If the Harary-Hill conjecture holds, then φs(k) ≤ √ 2 k3/4(1 + o(1)) Note: This is indeed very close to the original question of Bruce Richter.

Alfaro-Arroyo-Dernar-M. cr(cone(G))

Questions?

Alfaro-Arroyo-Dernar-M. cr(cone(G))