Graph coloring with no large monochromatic components

Nathan Linial*, Jiří Matoušek, Or Sheffet, Gábor Tardos

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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 languageAmerican English
Pages (from-to)115-122
Number of pages8
JournalElectronic Notes in Discrete Mathematics
Issue numberSPEC. ISS.
StatePublished - 15 Aug 2007


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