Graph coloring with no large monochromatic components

For a graph G and an integer t we let mcct(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(n2/(2t−1) ). It is also interesting to consider graphs of bounded degrees. Haxell, Szab´o, 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...

• Type: Journal
• Year: 2007
• Where: ENDM
• Authors: Nathan Linial, Jirí Matousek, Or Sheffet, Gábor Tardos
• Keywords: ENDM 2007, Monochromatic Connected Subgraph, N-vertex Graphs, Nontrivial Minor-closed Family