Coloring the edges of a random graph without a monochromatic giant component Reto Spöhel (joint...
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Coloring the edges of a random graph without a monochromatic giant component Reto Spöhel(joint with Angelika Steger and Henning Thomas)
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Definitions
Gn,m: graph drawn uniformly at random (u.a.r.) from all graphson n vertices with m edges.
With high probability (whp.): with probability tending to 1as n 1.
(Sharp) threshold for some property P:Function m0(n) such that
Example: Connectivity has a sharp threshold atm0(n) = n log n / 2
In this talk: all thresholds are of form m0(n) = c0n for some constant c0 > 0.
(n)
Whp. Gn,m does not satisfy P
if m(n) < (1 – ²) m0(n)
Whp. Gn,m satisfies P if m(n) > (1 + ²) m0(n)
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Phase Transition of the Random Graph
[Erdős, Rényi (1960)]The random graph Gn,cn whp. consists of
Giantc < 0.5c > 0.5
- components of size at most O(log n) if c < 0.5
- a single ‚giant‘ component of size £(n) and other components of size O(log n)
if c > 0.5
(n)
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Achlioptas Process
Random graph process: Edges appear u.a.r. one by one whp. giant component emerges after about n/2 steps
Achlioptas process: In every step get two random edges select one for inclusion in the graph and discard the
other one ) freedom of choice!
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Achlioptas Process
[Bohman, Frieze (2001)], ..., [Spencer, Wormald (2007)]In the Achlioptas process the emergence of the giant component can be slowed down or accelerated by a constant factor.
No exact thresholds are known; current best bounds are:[Spencer, Wormald (2007)]: Whp. a giant component can be avoided for at least 0.829n edge pairs, created within 0.334n edge pairs.
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Corresponding Offline Problem
Given n vertices and cn random edge pairs, is it possible to select one edge from every pair such that in the resulting graph every component has size o(n)?
[Bohman, Kim (2006)]This property has a threshold at c1n for some analytically computable constant c1 ¼ 0.9768.
Unrestricted variant ([Bohman, Frieze, Wormald (2004)]):Given n vertices and 2cn random edges, is it possible to select cn edges such that in the resulting graph every component has size o(n)?
This property has a (slightly higher!) threshold at 2c2n for some analytically computable constant c2 ¼ 0.9792.
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Coloring Variant of the Problem
Given n vertices and cn random edge pairs,is it possible to find a valid 2-edge-coloring such that every monochromatic component has size o(n)? Valid: Both colors are used exactly once in every
edge pair.
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Coloring Variant of the Problem
Let r ¸ 2 be fixed. Given n vertices and cn random r-sets of edges, is it possible to find a valid r-edge-coloring such that every monochromatic component has size o(n)? Valid: Each of the r colors is used exactly once in
every r-set.
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Coloring Variant of the Problem
Let r ¸ 2 be fixed. Given n vertices and cn random r-sets of edges, is it possible to find a valid r-edge-coloring such that every monochromatic component has size o(n)? Valid: Each of the r colors is used exactly once in
every r-set.
r = 4
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Coloring Variant of the Problem
Let r ¸ 2 be fixed. Given n vertices and cn random r-sets of edges, is it possible to find a valid r-edge-coloring such that every monochromatic component has size o(n)? Valid: Each of the r colors is used exactly once in
every r-set. Theorem [S., Steger, Thomas (2009+)]
For every r ¸ 2 this property has a threshold at for some analytically computable constant .
The threshold coincides with the threshold for r-orientability of the random graph Gn,rcn.
Unrestricted variant (ind. [Bohman, Frieze, Krivelevich, Loh, Sudakov]):Given n vertices and rcn random edges, is it possible to find an r-edge-coloring such that every monochromatic component has size o(n)?
This property has the same threshold as the restricted variant!
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
r-orientability
G is r-orientable if its edges can be oriented in such a way thatthe in-degree of every vertex is at most r.
In fact, G is r-orientable iff m(G) · r, wherem(G) := maxHµG e(H)=v(H) is the max. edge density of G.
The threshold for r-orientability of the random graph Gn,m was determined by [Fernholz, Ramachandran (SODA 07)] and independently by [Cain, Sanders, Wormald (SODA 07)].
Setting m = rcn the threshold is at .
r 2 3 4 5 6 7 8 9
0.882
0.959
0.980
0.989
0.994
0.996
0.998
0.999
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Upper Bound Proof
Let c > . Need to show: Whp. every valid r-edge-coloring of cn random r-sets of edges contains a monochromatic giant.
We sample edges without replacement. ) G := “ r-sets” is distributed like Gn,rcn
Density Lemma ([Bohman, Frieze, Wormald (2004)])Whp. all subgraphs in G of edge density ¸ 1+² have linear size.
Whp. we have m(G) ¸ (1+²)r ) 9 subgraph with edge density ¸ (1+²)r ) Every r-edge-coloring of G contains a monochromatic
(connected!) subgraph with edge density ¸ 1+².
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Lower Bound Proof - Idea
Let c < . Need to show: Whp. there exists a valid r-edge-coloring of cn random r-sets of edges in which every monochromatic component has size o(n).
“Inverse Two Round Exposure”: We generate cn random r-sets by first generating
(c+²)n random r-sets (with c+² < ) and then deleting ²n random r-sets.
Let G+ be the union of the (c+²)n r-sets (distributed like Gn,r(c+²)n).
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Lower Bound Proof - Outline
How to use this idea (borrowed from [Bohman, Kim (2006)]): First Round: Find a valid r-edge-coloring of G+ in which
every monochromatic component is low-connected (at most unicyclic)
Second Round: Show that the edge deletion breaks the low-connected components into small ones.
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Lower Bound – First Round
Fact: The chromatic index of a bipartite graph G equals ¢(G)
This yields a valid r-edge-coloring of E(G+) such that in every color class every vertex has in-degree at most 1.
) Every monochromatic component is unicyclic or a tree.
2
1
5
3 4
2
1
5
3 4
1
2
3
4
5
B
G+
V(G+) r-setsEvery edge- belongs to one r-set- points to one vertex
1
2
3
4
5¢(B) = r
2
1
5
3 4
r = 2
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Lower Bound – Second Round (Sketch)
Consider a fixed color class with components C1+, …,
Cs+
Remove one edge from every cycle Lemma: Deleting ²n random r-sets breaks the
resulting trees into components of size o(n). Then: Every component Ci
+ breaks into components of size at most 2o(n) = o(n).
Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component
EuroComb 2009
Summary
Avoiding monochromatic giants in r-edge-colorings of random graphs has the same threshold as r-orientability of random graphs.
No difference between restricted and unrestricted setting (in contrast to edge-selection problems)
Related Work Online setup Creating giants
Open Questions Vertex-Coloring
Thank you!
[Bohman, Frieze, Krivelevich, Loh, Sudakov (2009+)]