Coloring kcolorable graphs using smaller palettes Eran Halperin

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 D using D 1 -2/k log 1/k. D colors )instead of D 1 -2/k log 1/2 D colors [KMS([

New coloring results Coloring k-colorable graphs using na(k) colors (instead of nb(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/a, then an independent set of size nf(a) 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 W(n 1 -a; ( then we can color any k-colorable graph using O(na) colors.

Coloring 3 -colorable graphs using O(n 1/2) colors [Wigderson[ A graph with maximum degree D can be easily colored using D+1 colors. If D < n 1/2, color using D+1 colors. Otherwise, let v be a vertex of degree D. Then, N(v) is 2 -colorable and contains an independent set of size D/2>= 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, …, vn such that if (i, j) in E then <vi, vj>=-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 Rn. Let. I’ is obtained from I by removing a vertex from each edge of I.

Constructing the sets I and I’

Analysis

Analysis (Cont(.

Analysis (Cont(.

Analysis (Cont(.

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.

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 Ti satisfies: | • Ti|=W(d 2/s( • Ti has an independent subset of size

A lemma of Blum

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

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 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/D 1 -2/k= n 1 -a(k. (

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.

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)>n 1 -a(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 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(d 2/s) and Ti 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[