Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs Henning Thomas (joint with Reto...

Avoiding Monochromatic Giants inEdge-Colorings of Random GraphsHenning Thomas(joint with Reto Spöhel, Angelika Steger)

Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 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 o(n) if c < 0.5

- a single giant component of size (n) and other components of size o(n)

if c > 0.5

(n)


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.


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.


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.



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.




once in every r-set.

r = 4




once in every r-set. Theorem [Spöhel, Steger, T.]

For every r ¸ 2 the property has a threshold at cr*n

for some analytically computable constant cr*.

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!


r-Orientability

G is r-orientable if there exists an orientation such that the 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 rcr*n.r 2 3 4 5 6 7 8 9

cr* 0.88

20.95

90.98

00.98

90.99

40.99

60.99

80.99

9


Upper Bound Proof

Let c > cr*. 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 valid r-edge-coloring of G contains a

monochromatic (connected!) subgraph with edge density ¸ 1+².


Lower Bound Proof - Idea

Let c < cr*. 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 < cr*) 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).


Lower Bound Proof - Outline

How to use this idea (inspired by [Bohman, Kim (2007)]): First Round: Find a valid r-edge-coloring of G+ in

which every monochromatic component is low-connected

Second Round: Show that the edge deletion breaks the low-connected components into small ones.


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


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).


Summary

Avoiding monochromatic giants in edge-colorings of random graphs has the same threshold as orientability of random graphs.

No difference between balanced and unbalanced setting (in contrast to edge-selection problems)

Related Work Online setup Creating the giant

Open Questions Vertex-Coloring

[Bohman, Frieze, Krivelevich, Loh, Sudakov]Po-Shen‘s Talk


Lower Bound – Second Round (Sketch)

A vertex ‘survives’ if its first log(n)/Kancestors are not deleted

Pr[u survives]·(1-²/(c+²))log(n)/K· n-²/(c+²)K

E[#surviving vertices]=O(n1-²/(c+²)K) Markov: Whp. #surviving vertices=o(n) (*) Lemma ([Bohman, Kim (2007)]):

Whp. all trees in Gn,rcn with depth atmost log(n)/K have o(n) vertices. (**)

Consider a tree after the edge deletion Conditioning on (*) and (**)

such a tree has o(n) vertices

log(n)/K

Tree of depth at most log(n)/K

Surviving vertices

Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs Henning Thomas (joint with Reto...

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