Coloring k -colorable graphs using smaller palettes

1. Coloring k-colorable graphsusing smaller palettes Eran Halperin Ram Nathaniel Uri Zwick Tel Aviv University

2. New coloring results Coloring k-colorable graphs of maximum degree Dusing D1-2/klog1/kDcolors (instead of D1-2/klog1/2Dcolors [KMS])

3. New coloring results Coloring k-colorable graphs using na(k)colors (instead of nb(k)colors [KMS])

4. An extension of Alon-Kahale AK: If a graph contains an independent set of size n/k+m, k integer, then an independent set of size m3/(k+1) can be found in polynomial time. Extension: If a graph contains an independent set of size n/a, then an independent set of size nf(a) can be found in polynomial time, where

5. Graph coloring basics If in any k-colorable graph on n vertices we can find, in polynomial time, one of Two vertices that have the same color under some valid k-coloring ; An independent set of size W(n1-a) ; then we can color any k-colorable graph using O(na) colors.

6. Coloring 3-colorable graphs using O(n1/2) colors [Wigderson] A graph with maximum degree D can be easily colored using D+1colors. If D <n1/2, color using D+1colors. Otherwise, let v be a vertex of degree D. Then, N(v) is 2-colorable and contains an independent set of size D/2>=n1/2/2.

7. Vector k-Coloring [KMS] Avector k-coloring of a graph G=(V,E) is a sequence of unit vectors v1,v2,…,vn such that if (i,j) in E then <vi,vj>=-1/(k-1).

8. Finding large independent sets Let G=(V,E) be a 3-colorable graph. Let r be a random normally distributed vector in Rn. Let . I’ is obtained from I by removing a vertex from each edge of I.

9. Constructing the sets I and I’

10. Analysis

11. Analysis (Cont.)

13. Analysis (Cont.)

14. Analysis (Cont.)

15. A simple observation Suppose G=(V,E) is k-colorable. Either G[N(u,v)] is (k-2)-colorable, or u and v get the same color under any a k-coloring of G.

16. A lemma of Blum Let G=(V,E) be a k-colorable graph with minimum degree d for every Then, it is possible to construct, in polynomial time, a collection {Ti} of about n subsets of V such that at least one Tisatisfies: |Ti|=W(d2/s) Tihas an independent subset of size

17. A lemma of Blum

18. Graph coloring techniques Karger Motwani Sudan Wigderson Blum Alon Kahale Blum Karger Our Algorithm

19. The new algorithm Step 0: If k=2, color the graph using 2 colors. If k=3, color the graph using n3/14 colors using the algorithm of Blum and Karger.

20. The new algorithm Step 1: Repeatedly remove from the graph vertices of degree at most na(k)/(1-2/k). Let U be the set of vertices removed, and W=V-U. Average degree of G[U] is at most na(k)/(1-2/k). Minimum degree of G[W] at least na(k)/(1-2/k). If |U|>n/2, use [KMS] to find an independent set of size n/D1-2/k= n1-a(k).

21. Step 1 Let d=na(k)/(1-2/k). Average degree of G[U] is at most d. Minimum degree of G[W] at least d.

22. The new algorithm Step 2: For every u,v such that N(u,v)>n(1-a(k)/(1-a(k-2)), apply the algorithm recursively on G[N(u,v)] and k-2. If G[N(u,v)] is (k-2)-colorable, we get an independent set of size |N(u,v)|1-a(k-2)>n1-a(k). Otherwise, we can infer*that u and v must be assigned the same color.

23. The new algorithm Step 3:If we reach this step then |W|>n/2, the minimum degree of G[W] is at least na(k)/(1-2/k), and for every u,v in W, N(u,v)>n(1-a(k)/(1-a(k-2)). By Blum’s lemma, we can find a collection {Ti} of about n subsets of W such that at least one Ti satisfies |Ti|=W(d2/s) and Tihas an independent subset of size . By the extension of the Alon-Kahale result, we can find an IS of size

24. The recurrence relation

25. Hardness results It is NP-hard to 4-color 3-colorable graphs [Khanna,Linial,Safra ‘93] [Guruswami,Khanna ‘00] For any k, it is NP-hard to k-color 2-colorable hypergraphs [Guruswami,Hastad,Sudan ‘00]