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

Motivation: Two conjectures and the cone ◮ Harary-Hill Conjecture: � 64 n ( n − 2) 2 ( n − 4) , 1 n is even; cr ( K n ) = 64 ( n − 1) 2 ( n − 3) 2 , 1 n is odd. ◮ Albertson’s Conjecture: χ ( G ) ≥ r ⇒ cr ( G ) ≥ cr ( K r ) Alfaro-Arroyo-Dernar-M. cr(cone(G))

Motivation: Two conjectures and the cone ◮ Harary-Hill Conjecture: � 64 n ( n − 2) 2 ( n − 4) , 1 n is even; cr ( K n ) = 64 ( n − 1) 2 ( n − 3) 2 , 1 n is odd. ◮ Albertson’s Conjecture: χ ( G ) ≥ r ⇒ cr ( G ) ≥ cr ( K r ) 1 In the H-H Conjecture: K n +1 = C ( K n ). 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 ( K r ) Alfaro-Arroyo-Dernar-M. cr(cone(G))

Motivation: Two conjectures and their variation Albertson’s Conjecture: χ ( G ) ≥ r ⇒ cr ( G ) ≥ cr ( K r ) A variation asked by Bruce Richter: Given n ≥ 5 and a graph G with cr ( G ) ≥ cr ( K n ), does it follow that cr ( CG ) ≥ cr ( K n +1 )? Alfaro-Arroyo-Dernar-M. cr(cone(G))

Richter’s question cr ( G ) ≥ cr ( K n ) ⇒ cr ( CG ) ≥ cr ( K n +1 )? Observation: True for n = 5. (Because C ( K 5 ) = K 6 and cr ( C ( K 3 , 3 )) = 3) Alfaro-Arroyo-Dernar-M. cr(cone(G))

Richter’s question cr ( G ) ≥ cr ( K n ) ⇒ cr ( CG ) ≥ cr ( K n +1 )? Observation: True for n = 5. (Because C ( K 5 ) = K 6 and cr ( C ( K 3 , 3 )) = 3) False for n = 6. cr ( G ) = 3 = cr ( K 6 ) and cr ( C ( G )) = 6 < cr ( K 7 ) = 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 ) ≤ 3 k Upper bound: Previous graph with edge-multiplicities r has crossing number 3 r 2 and its cone has crossing number 3 r 2 + 3 r . Alfaro-Arroyo-Dernar-M. cr(cone(G))

The lower bound � 1 Theorem. cr ( C ( G )) ≥ cr ( G ) + 2 cr ( G ) Proof. Step 1: Obtain a 1-page drawing of G : e 3 a e 4 e 2 e 5 e 1 � e 6 D D 0 D (a) (b) (c) Alfaro-Arroyo-Dernar-M. cr(cone(G))

The lower bound � 1 Theorem. cr ( C ( G )) ≥ cr ( G ) + 2 cr ( 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 1 8 m + 1 64 − 1 8 > 1 2 m + 2 m 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 1 8 k + 1 64 + 1 2 k − 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 � 1 Theorem. cr ( C ( G )) ≥ cr ( G ) + 2 cr ( G ) Proof (cont’d). D 0 If too few crossings, we obtain a drawing of G with < cr ( G ) crossings. � Alfaro-Arroyo-Dernar-M. cr(cone(G))

Simple graphs φ s ( k ) := min { cr ( C ( G )) − cr ( G ) | cr ( G ) = k , G is simple } � The lower bound φ s ( k ) ≥ k / 2 still holds. Theorem. φ s (3) = 3 , φ s (4) = 4 , φ s (5) = 5. Obvious conjecture! Alfaro-Arroyo-Dernar-M. cr(cone(G))

Simple graphs φ s ( k ) := min { cr ( C ( G )) − cr ( G ) | cr ( G ) = k , G is simple } � The lower bound φ s ( k ) ≥ k / 2 still holds. Theorem. φ s (3) = 3 , φ s (4) = 4 , φ s (5) = 5. Obvious conjecture! Theorem. φ s ( k ) = O ( k 3 / 4 ) Alfaro-Arroyo-Dernar-M. cr(cone(G))

Conjecture √ 2 k 3 / 4 (1 + o (1)) Conjecture φ s ( k ) = Alfaro-Arroyo-Dernar-M. cr(cone(G))

Conjecture √ 2 k 3 / 4 (1 + o (1)) Conjecture φ s ( k ) = This specific form of the conjecture is due to the following observation: Proposition If the Harary-Hill conjecture holds, then √ 2 k 3 / 4 (1 + o (1)) φ s ( k ) ≤ 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))