#### Title

Highly Connected Monochromatic Subgraphs of Multicoloured Graphs

#### Document Type

Article

#### Publication Date

2009

#### Disciplines

Mathematics

#### Abstract

We consider the following question of Bollobás: given an *r*-coloring of *E*(*K** _{n}*), how large a

*k*-connected subgraph can we find using at most

*s*colors? We provide a partial solution to this problem when

*s*=1 (and

*n*is not too small), showing that when

*r*=2 the answer is

*n*−2

*k*+2, when

*r*=3 the answer is ⌊(

*n*−

*k*)/2⌋+1 or ⌈(

*n*−

*k*)/2⌉+1, and when

*r*−1 is a prime power then the answer lies between

*n*/(

*r*−1)−11(

*k*

^{2}−

*k*)

*r*and (

*n*−

*k*+1)/(

*r*−1)+

*r*. The case

*s*≥2 is considered in a subsequent paper (Liu et al.[6]), where we also discuss some of the more glaring open problems relating to this question.

#### Recommended Citation

Liu, H., Morris, R., Prince, N. (2009). Highly connected monochromatic subgraphs of multicoloured graphs. *Journal of Graph Theory,* 61(1), 22-44.

## Comments

At the time of publication, Noah Prince was affiliated with the University of Illinois at Urbana-Champaign.