Graph coloring with no large monochromatic components

Nathan Linial, Jiří Matoušek, Or Sheffet, G. Tardos

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

For a graph G and an integer t we let m c ct (G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that mcc2 (G) = O (n2 / 3) for any n-vertex graph G ∈ F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F, 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 (n(2 / 2 t - 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) ≥ cε n. 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 m c ct (G).

Original languageEnglish
Pages (from-to)115-122
Number of pages8
JournalElectronic Notes in Discrete Mathematics
Volume29
Issue numberSPEC. ISS.
DOIs
Publication statusPublished - Aug 15 2007

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Graph Coloring
Coloring
Color
Graph in graph theory
Regular Graph
Maximum Degree
Subgraph
Minor
Vertex of a graph
Asymptotically Optimal
Planar graph
Colouring
Closed
Integer

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

Graph coloring with no large monochromatic components. / Linial, Nathan; Matoušek, Jiří; Sheffet, Or; Tardos, G.

In: Electronic Notes in Discrete Mathematics, Vol. 29, No. SPEC. ISS., 15.08.2007, p. 115-122.

Research output: Contribution to journalArticle

Linial, Nathan ; Matoušek, Jiří ; Sheffet, Or ; Tardos, G. / Graph coloring with no large monochromatic components. In: Electronic Notes in Discrete Mathematics. 2007 ; Vol. 29, No. SPEC. ISS. pp. 115-122.
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