#### Title

### Forbidden Properly Edge-Colored Subgraphs that Force Large Highly Connected Monochromatic Subgraphs

#### Document Type

Article

#### Publication Date

7-2017

#### Publication Title

Graphs and Combinatorics

#### DOI

10.1007/s00373-017-1804-5

#### ISSN

1435-5914

#### Abstract

We consider the connected graphs G that satisfy the following property: If *n*≫*m*≫*k* are integers, then any coloring of the edges of *K _{n}*, using

*m*colors, containing no properly colored copy of

*G*, contains a monochromatic

*k*-connected subgraph of order at least

*n−f*(

*G,k,m*) where

*f*does not depend on

*n*. If we let

*G*denote the set of graphs satisfying this statement, we exhibit some infinite families of graphs in

*G*as well as conjecture that the cycles in

*G*are precisely those whose lengths are divisible by 3. Our main result is that

*C*∈

_{6}*G*.

#### Recommended Citation

Katic, Robert, Colton Magnant, Pouria Salehi Nowbandegani.
2017.
"Forbidden Properly Edge-Colored Subgraphs that Force Large Highly Connected Monochromatic Subgraphs."
*Graphs and Combinatorics*, 33 (4): 969-979.
doi: 10.1007/s00373-017-1804-5

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/592