avoiding monochromatic giants in edge-colorings of random graphs henning thomas (joint with reto...
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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)
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 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.
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 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.
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 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.
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 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.
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 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
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 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 [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!
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 2009
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
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 2009
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.
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 2009
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).
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 2009
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.
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 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
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 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).
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 2009
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
Henning Thomas Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs ETH Zurich 2009
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