Graph colouring with no large monochromatic components

  • Authors:

  • Nathan Linial;JiŘ/Í/ MatouŠ/ek;Or Sheffet;GÁ/bor Tardos

  • Affiliations:

  • School of computer science and engineering, hebrew university, jerusalem, israel (e-mail: nati@cs.huji.ac.il/ or.sheffet@gmail.com);Department of applied mathematics and institute of theoretical computer science (iti), charles university, malostranské/ ná/m. 25, 118 00 praha 1, czech republic (e-mail: matousek@kam.mff. ...;School of computer science and engineering, hebrew university, jerusalem, israel (e-mail: nati@cs.huji.ac.il/ or.sheffet@gmail.com);School of computing science, simon fraser university, burnaby, bc, canada and ré/nyi institute, budapest, hungary (e-mail: tardos@cs.sfu.ca)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2008

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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 $\sqrt n$ and n. We also offer a Ramsey-theoretic perspective of the quantity mcct(G).