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Better coloring of 3-colorable graphs

Ken-ichi Kawarabayashi, Mikkel Thorup, Hirotaka Yoneda

TL;DR

A new combinatorial algorithm for 3-colorable graphs with minimum degree >√n that can make progress towards a k-coloring for some k=√n/· no(1) and shows that (n0.19747) colors suffice for coloring 3-colorable graphs.

Abstract

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. This is one of the most challenging problems in graph algorithms. In this paper using Blum's notion of ``progress'', we develop a new combinatorial algorithm for the following: Given any 3-colorable graph with minimum degree $\ds>\sqrt n$, we can, in polynomial time, make progress towards a $k$-coloring for some $k=\sqrt{n/\ds}\cdot n^{o(1)}$. We balance our main result with the best-known semi-definite(SDP) approach which we use for degrees below $n^{0.605073}$. As a result, we show that $\tO(n^{0.19747})$ colors suffice for coloring 3-colorable graphs. This improves on the previous best bound of $\tO(n^{0.19996})$ by Kawarabayashi and Thorup in 2017.

Better coloring of 3-colorable graphs

TL;DR

A new combinatorial algorithm for 3-colorable graphs with minimum degree >√n that can make progress towards a k-coloring for some k=√n/· no(1) and shows that (n0.19747) colors suffice for coloring 3-colorable graphs.

Abstract

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. This is one of the most challenging problems in graph algorithms. In this paper using Blum's notion of ``progress'', we develop a new combinatorial algorithm for the following: Given any 3-colorable graph with minimum degree , we can, in polynomial time, make progress towards a -coloring for some . We balance our main result with the best-known semi-definite(SDP) approach which we use for degrees below . As a result, we show that colors suffice for coloring 3-colorable graphs. This improves on the previous best bound of by Kawarabayashi and Thorup in 2017.
Paper Structure (33 sections, 17 theorems, 39 equations, 1 figure, 4 algorithms)

This paper contains 33 sections, 17 theorems, 39 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Interplay between combinatorial and SDP approaches and our main result
  3. Purely combinatorial bounds
  4. Combination with SDP
  5. Combinatorial algorithm for balance with SDP
  6. Our result
  7. Techniques
  8. Preliminaries
  9. Recursive combinatorial coloring
  10. Inner loop
  11. Cut-or-color
  12. $\bm X$-extension
  13. $\bm{Y}$-extension
  14. Recursion towards a monochromatic set
  15. Regularization
  16. ...and 18 more sections

Key Result

Theorem 1

In polynomial time, for any 3-colorable graph with $n$ vertices with minimum degree $\Delta>n^{0.5}$, we can make progress towards a $k$-coloring for some $k=n^{o(1)} (n/\Delta)^{1/2}$

Figures (1)

  • Figure 1: Side cuts and sparse cuts

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3: KT17
  • Lemma 4: Blum94
  • Lemma 5: Blum94
  • Lemma 6: KT17
  • Lemma 7: KT17
  • Theorem 8: Chl07
  • Lemma 9
  • Lemma 10
  • ...and 7 more