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Graph Colouring with No Large Monochromatic Components

Published online by Cambridge University Press:  01 July 2008

NATHAN LINIAL
Affiliation:
School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel (e-mail: nati@cs.huji.ac.il; or.sheffet@gmail.com)
JIŘÍ MATOUŠEK
Affiliation:
Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI), Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic (e-mail: matousek@kam.mff.cuni.cz)
OR SHEFFET
Affiliation:
School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel (e-mail: nati@cs.huji.ac.il; or.sheffet@gmail.com)
GÁBOR TARDOS
Affiliation:
School of Computing Science, Simon Fraser University, Burnaby, BC, Canada and Rényi Institute, Budapest, Hungary (e-mail: tardos@cs.sfu.ca)

Abstract

For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let be any non-trivial minor-closed family of graphs. We show that mcc2(G) = O(n2/3) for any n-vertex graph G. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such , and every fixed t we show that mcct(G)=O(n2/(t+1)). On the other hand, we have examples of graphs G with no Kt+3 minor and with mcct(G)=Ω(n2/(2t−1)).

It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos proved mcc2(G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc2(G)=Ω(n), and more sharply, for every ϵ > 0 there exists cϵ > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ϵ for all subgraphs, and with mcc2(G) ≥ cϵn. For 6-regular graphs it is known only that the maximum order of magnitude of mcc2 is between and n.

We also offer a Ramsey-theoretic perspective of the quantity mcct(G).

Type
Paper
Copyright
Copyright © Cambridge University Press 2008

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