Coloring k-colorable graphs using smaller palettes

We obtain the following new coloring results:<ul><li>A 3-colorable graph on <i>n</i> vertices with maximum degree Δ can be colored, in polynomial time, using <i>&Ogr;</i>((Δ log Δ)<sup>1/3</sup> ·log <i>n</i>) colors. This slightly improves an <i>&Ogr;</i>((Δ<sup>1/3</sup> log<sup>½</sup> Δ) · log <i>n</i>) bound given by Karger, Motwani and Sudan. More generally, <i>k</i>-colorable graphs with maximum degree Δ can be colored, in polynomial time, using <i>&Ogr;</i>((Δ<sup>1-2/<i>k</i></sup> log<sup>1/<i>k</i></sup> Δ) · log <i>n</i>) colors. </li><li>A 4-colorable graph on <i>n</i> vertices can be colored, in polynomial time, using <i>&Ogr;</i>(<i>n</i><sup>7/19</sup>) colors. This improves an <i>&Ogr;</i>(<i>n</i><sup>2/5</sup>) bound given again by Karger, Motwani and Sudan. More generally, <i>k</i>-colorable graphs on <i>n</i>-vertices can be colored, in polynomial time, using <i>&Ogr;</i>(<i>n</i><sup>αk</sup>) colors, where α<subscrpt>5</subscrpt> = 97/207, α<subscrpt>6</subscrpt> = 43/79, α<subscrpt>7</subscrpt> = 1391/2315, α<subscrpt>8</subscrpt> = 175/271, … </li></ul> The first result is obtained by a slightly more refined probabilistic analysis of the semidefinite programming based coloring algorithm of Karger, Motwani and Sudan. The second result is obtained by combining the coloring algorithm of Karger, Motwani and Sudan, the combinatorial coloring algorithms of Blum and an extension of a technique of Alon and Kahale (which is based on the Karger, Motwani and Sudan algorithm) for finding relatively large independent sets in graphs that are guaranteed to have very large independent sets. The extension of the Alon and Kahale result may be of independent interest.

[1]  Venkatesan Guruswami,et al.  On the hardness of 4-coloring a 3-collorable graph , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[2]  Benny Sudakov,et al.  Approximate coloring of uniform hypergraphs , 1998, J. Algorithms.

[3]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[4]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[5]  David R. Karger,et al.  Approximate graph coloring by semidefinite programming , 1998, JACM.

[6]  Sanjeev Mahajan,et al.  Derandomizing Approximation Algorithms Based on Semidefinite Programming , 1999, SIAM J. Comput..

[7]  David R. Karger,et al.  An Õ(n^{3/14})-Coloring Algorithm for 3-Colorable Graphs , 1997, Information Processing Letters.

[8]  Nathan Linial,et al.  On the Hardness of Approximating the Chromatic Number , 1993, [1993] The 2nd Israel Symposium on Theory and Computing Systems.

[9]  Magnús M. Halldórsson,et al.  A Still Better Performance Guarantee for Approximate Graph Coloring , 1993, Information Processing Letters.

[10]  Uriel Feige,et al.  Zero Knowledge and the Chromatic Number , 1998, J. Comput. Syst. Sci..

[11]  Magnús M. Hallórsson.  A still better performance guarantee for approximate graph coloring , 1993 .

[12]  Noga Alon,et al.  Approximating the independence number via theϑ-function , 1998, Math. Program..

[13]  Avi Wigderson,et al.  Improving the performance guarantee for approximate graph coloring , 1983, JACM.

[14]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[15]  Avrim Blum,et al.  New approximation algorithms for graph coloring , 1994, JACM.

[16]  Nathan Linial,et al.  On the Hardness of Approximating the Chromatic Number , 2000, Comb..