1. Coloring the edges of a random graph without a monochromatic giant component Reto Spöhel (joint with Angelika Steger and Henning Thomas) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A

2. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 2009 Definitions  G n, m : graph drawn uniformly at random (u.a.r.) from all graphs on n vertices with m edges.  With high probability (whp.): with probability tending to 1 as n  1.  (Sharp) threshold for some property P : Function m 0 ( n ) such that  Example: Connectivity has a sharp threshold at m 0 ( n ) = n log n / 2  In this talk: all thresholds are of form m 0 ( n ) = c 0 n for some constant c 0 > 0. Whp. G n, m does not satisfy P if m ( n ) < (1 – ² ) m 0 ( n ) Whp. G n, m satisfies P if m ( n ) > (1 + ² ) m 0 ( n )

3. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 2009 Phase Transition of the Random Graph  [Erd ő s, Rényi (1960)] The random graph G n, cn whp. consists of Giant c < 0.5 c > 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

4. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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!

5. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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 0.829 n edge pairs,  created within 0.334 n edge pairs.

6. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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 c 1 n for some analytically computable constant c 1 ¼ 0.9768. Unrestricted variant ([Bohman, Frieze, Wormald (2004)]): Given n vertices and 2 cn 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 2 c 2 n for some analytically computable constant c 2 ¼ 0.9792.

7. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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.

8. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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.

9. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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

10. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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 G n, 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!

11. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 2009 r -orientability  G is r -orientable if its edges can be oriented in such a way that the in-degree of every vertex is at most r.  In fact, G is r -orientable iff m ( G ) · r, where m ( G ) := max H µ G e ( H ) =v ( H ) is the max. edge density of G.  The threshold for r -orientability of the random graph G n, m was determined by [Fernholz, Ramachandran (SODA 07)] and independently by [Cain, Sanders, Wormald (SODA 07)].  Setting m = rcn the threshold is at. r 23456789 0.8820.9590.9800.9890.9940.9960.9980.999

12. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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 G n, 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.

13. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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 G n, r ( c + ² ) n ).

14. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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.

15. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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 34 2 1 5 34 1 2 3 4 5 B G+G+ V(G+)V(G+) r -sets Every edge - belongs to one r -set - points to one vertex 1 2 3 4 5 ¢(B) = r¢(B) = r 2 1 5 34 r = 2

16. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 2009 Lower Bound – Second Round (Sketch)  Consider a fixed color class with components C 1 +, …, C s +  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 C i + breaks into components of size at most 2 o ( n ) = o ( n ).

17. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 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+)]

18. Reto SpöhelColoring the edges of a random graph without a monochromatic giant componentEuroComb 2009