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+)]

Reto Spöhel Coloring the edges of a random graph without a monochromatic giant component

EuroComb 2009