Coloring k-colorable graphs using smaller palettes Eran Halperin Ram Nathaniel Uri Zwick Tel Aviv University

New coloring results Coloring k-colorable graphs of maximum degree  using   -2/k  log 1/k  colors (instead of   -2/k  log 1/2  colors [KMS])

New coloring results Coloring k-colorable graphs using n  (k) colors (instead of n  (k)  colors [KMS])

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 m 3/(k+1) can be found in polynomial time. Extension: If a graph contains an independent set of size n  then an independent set of size n f(  can be found in polynomial time, where

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  (n 1-  ) ; then we can color any k-colorable graph using O(n  ) colors.

Coloring 3-colorable graphs using O(n 1/2 ) colors [Wigderson] A graph with maximum degree  can be easily colored using  colors. If  <  n 1/2, color using  colors. Otherwise, let v be a vertex of degree   hen, N(v) is 2-colorable and contains an independent set of size  n 1/2 /2 

Vector k-Coloring [KMS] A vector k-coloring of a graph G=(V,E) is a sequence of unit vectors v 1,v 2,…,v n such that if (i,j) in E then =-1/(k-1).

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

Constructing the sets I and I’

Analysis

Analysis (Cont.)

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

A lemma of Blum Let G=(V,E) be a k-colorable graph with minimum degree   for every Then, it is possible to construct, in polynomial time, a collection {T i } of about n subsets of V such that at least one T i satisfies: |T i |=    s) T i has an independent subset of size

A lemma of Blum

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

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

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

Step 1 Average degree of G[U] is at most . Minimum degree of G[W] at least . Let  n  (k)/(1-2/k).

The new algorithm Step 2: For every u,v such that N(u,v)>n (1-  (k)/(1-  (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-  (k-2) >n 1-  (k). Otherwise, we can infer * that u and v must be assigned the same color.

The new algorithm Step 3: If we reach this step then |W|>n/2, the minimum degree of G[W] is at least n  (k)/(1-2/k), and for every u,v in W, N(u,v)>n (1-  (k)/(1-  (k-2)). By Blum’s lemma, we can find a collection {T i } of about n subsets of W such that at least one T i satisfies |T i |=    s) and T i has an independent subset of size. By the extension of the Alon-Kahale result, we can find an IS of size

The recurrence relation

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]