Graph coloring with no large monochromatic components
Abstract
For a graph G and an integer t we let mcc_t(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 minorclosed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any nvertex graph G \in 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 mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t1)}). It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5. We show that there are nvertex 7regular graphs G with \mcc_2(G)=\Omega(n), and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and nvertex graphs of maximum degree 7, average degree at most 6+\epsilon for all subgraphs, and with mcc_2(G)\ge c_\eps n. For 6regular graphs it is known only that the maximum order of magnitude of \mcc_2 is between \sqrt n and n. We also offer a Ramseytheoretic perspective of the quantity \mcc_t(G).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2007
 DOI:
 10.48550/arXiv.math/0703362
 arXiv:
 arXiv:math/0703362
 Bibcode:
 2007math......3362L
 Keywords:

 Mathematics  Combinatorics;
 05C15
 EPrint:
 13 pages, 2 figures