SLIDE 1 Hamilton cycles in the random geometric graph
Nick Wormald
University of Waterloo
SLIDE 2 Hamilton cycles in the random geometric graph
Nick Wormald
University of Waterloo
joint work with Tobias M¨ uller and ∗Xavier P´ erez Gim´ enez (∗also contributed to presentation)
SLIDE 3
Wireless networks
SLIDE 4
Random geometric graph
(Gilbert 1961) n vertices radius r = r(n) n → ∞
SLIDE 5
Random process: 0 ≤ r ≤ √ 2
SLIDE 6
Random process: 0 ≤ r ≤ √ 2
SLIDE 7
Random process: 0 ≤ r ≤ √ 2
SLIDE 8
Random process: 0 ≤ r ≤ √ 2
no giant component yet
SLIDE 9 Random process: 0 ≤ r ≤ √ 2
r ∼
giant component!
SLIDE 10
Random process: 0 ≤ r ≤ √ 2
still disconnected!
SLIDE 11 Random process: 0 ≤ r ≤ √ 2
connected = no isolated vertices (a.a.s.) r =
πn
SLIDE 12 Random process: 0 ≤ r ≤ √ 2
2-connected = no deg. 1 vertices (a.a.s.) r =
πn
SLIDE 13
Random process: 0 ≤ r ≤ √ 2
higher connectivity
SLIDE 14
Random process: 0 ≤ r ≤ √ 2
still large diameter: Θ(1/r) bad expansion
SLIDE 15
What about hamilton cycles?
SLIDE 16
What about hamilton cycles?
SLIDE 17 What about hamilton cycles?
Necessary conditions:
2-connected
SLIDE 18 What about hamilton cycles?
Necessary conditions:
2-connected Are they sufficient for the RGG?
SLIDE 19
Hamilton cycles in random graphs
G(n, m) is the random graph with n vertices and m edges chosen randomly ... ... a snapshot of the random graph process at time m. Thm (Bollob´ as 1984) Asymptotically almost surely, the first edge to give the graph min degree 2 also gives it a Hamilton cycle. Thm (Bollob´ as and Frieze 1985) Asymptotically almost surely, the first edge to give the graph min degree k also gives it k/2 edge-disjoint Hamilton cycles.
SLIDE 20 Proof technique for random graphs
Based on P´
SLIDE 21 Proof technique for random graphs
Based on P´
SLIDE 22 Proof technique for random graphs
Based on P´
SLIDE 23 Proof technique for random graphs
Based on P´
SLIDE 24
Hamilton cycles in random regular graphs
Gn,d: d-regular graph on n vertices chosen uniformly at random.
SLIDE 25
Hamilton cycles in random regular graphs
Let Yn be number of Hamilton cycles in Gn,3. Then EYn ∼ e π 2n 4 3 n/2 . Density of Yn/EYn:
SLIDE 26 Earlier results on RGG
In RGG, edges are added in increasing length. Thm (Penrose 1999) Asymptotically almost surely, the edge making the RGG have minimum degree k also makes it k-connected, and this happens for r ∼
Thm (Petit 2001) The RGG with r =
- ω(log n)/n a.a.s. has a Hamilton cycle.
Thm (D´ ıaz, Mitsche & P´ erez Gim´ enez 2007) For any ǫ > 0, the RGG with r ≥ (1 + ǫ)
πn
a.a.s. has a Hamilton cycle. (And extensions to general ℓp norm.)
SLIDE 27
Recent results
Thm (Balogh, Bollob´ as, Krivelevich, M¨ uller, P´ erez Gim´ enez, Walters & W. 2010) In the RGG process: Hamiltonian ⇐ ⇒ min. deg. ≥ 2 (a.a.s.) (extension to general dimension and ℓp norm) Thm (Balogh, Bollob´ as & Walters 2010) Weaker analogue for the k-Nearest Neighbour Graph. Thm (Krivelevich & M¨ uller 2010) Pancyclic ⇐ ⇒ min. deg. ≥ 2 (a.a.s.)
SLIDE 28
Recent results
Thm (Balogh, Bollob´ as, Krivelevich, M¨ uller, P´ erez Gim´ enez, Walters & W. 2010) In the RGG process: Hamiltonian ⇐ ⇒ min. deg. ≥ 2 (a.a.s.) (extension to general dimension and ℓp norm) Thm (M¨ uller, P´ erez Gim´ enez & W. 2010) k/2 disjoint Hamilton cycles ⇐ ⇒ min. deg. ≥ k (a.a.s.) (extension to general dimension and ℓp norm) For k odd there is an additional disjoint perfect matching.
SLIDE 29
Proof for disjoint Hamilton cycles
SLIDE 30 Preliminaries
From Penrose (2003): Let rk be the smallest r such that RGG is k-connected. Then πnr 2
k − log n − (2k − 3) log log n
is bounded in probability. Relevant r: r =
πn
SLIDE 31 First step: tesselation
r =
πn
δr dense (≥ M points) sparse (< M points)
SLIDE 32 First step: tesselation
r =
πn
δr dense (≥ M points) sparse (< M points)
SLIDE 33 First step: tesselation
r =
πn
δr dense (≥ M points) sparse (< M points)
bad cells
SLIDE 34 Simple computations:
P(a given cell is sparse) =
M−1
n t
= (Θ(δ2 log n))M−1(log n)O(1)n−πδ2 ... after some computations ... Every bad component is “small”.
SLIDE 35
Hamilton cycles: large-scale template
SLIDE 36
Hamilton cycles: large-scale template
SLIDE 37
Hamilton cycles: large-scale template
SLIDE 38
Rerouting at dense cells
SLIDE 39
Rerouting at dense cells
SLIDE 40
Extension into sparse good cells
SLIDE 41
Extension into sparse good cells
SLIDE 42
Extension into sparse good cells
SLIDE 43
Extension into sparse good cells
SLIDE 44
Extension into bad cells
a lot harder!
SLIDE 45
k = 4
SLIDE 46
k = 4
SLIDE 47
k = 4 But not 4-connected.
SLIDE 48
First solution for bad cells
Let graph G consist of a clique on vertex set J, |J| = j, and a bipartite graph H with parts J and B, where each vertex in J has degree at least k; for each v, v′ ∈ J, |NG(v) ∪ NG(v′) \ {u, v}| ≥ k; some vertex in J has degree at least k + 1. Then G contains a packing of k/2 edge-disjoint linear forests, with each vertex in J of degree 2 in each forest.
SLIDE 49
k = 6
SLIDE 50
k = 6
SLIDE 51
k = 6 [Conjecture that ‘degree ≥ k + 1’ condition unnecessary.]
SLIDE 52
Second solution for bad cells
For sufficiently small η > 0 and relevant r, every set of j ≥ 2 vertices in a circle of radius ηr (satisfying a certain max degree condition) has k common neighbours.
SLIDE 53 Open question
What if k is not fixed? In particular: Are there a.a.s. ⌊ δ(RGG)
2
⌋ edge disjoint Hamilton cycles?
