# "Distinguishing Chromatic Numbers of Bipartite Graphs"

## Document Type

Presentation Abstract

## Presentation Date

5-19-2010

## Abstract

Graph colouring problems have a long history and many variations. The classic *graph colouring problem* is to assign colours to the vertices of a graph *G* so that adjacent vertices receive different colours, and so that the total number of colours used is minimum. This minimum is the *chromatic number* of *G*, denoted *χ*(G).

In a 2006 article, Karen Collins and Ann Trenk introduce a variation of the chromatic number, called the *distinguishing chromatic number*. A colouring of the vertices of a graph *G* is *distinguishing* provided no automorphism of *G*, other than the identity, preserves the colours of the vertices. The distinguishing chromatic number of *G*, *X _{D}*(

*G*), is the minimum number of colours required to colour the vertices of

*Gso*that the resulting colouring is distinguishing. In their article, Collins and Trenk prove an analogue of Brooks' Theorem for

*X*

_{D}: if

*G*is a connected graph with maximum degree Δ, then

*X*(

_{D}*G*) ≤ 2Δ, with equality if and only if

*G*is

*K*

_{Δ,Δ}or a cycle on six vertices.

In this talk, I will outline what is known about the distinguishing chromatic number. I will also describe some joint work with Claude Laflamme, in which we restrict our attention to bipartite graphs, obtaining a slight improvement to the result of Collins and Trenk, and disproving one of their conjectures.

## Recommended Citation

Seyffarth, Karen, ""Distinguishing Chromatic Numbers of Bipartite Graphs"" (2010). *Colloquia of the Department of Mathematical Sciences*. 354.

https://scholarworks.umt.edu/mathcolloquia/354

## Additional Details

Wednesday, 19 May 2010

2:10 p.m. in Math 103

3:00 p.m. Refreshments in Math Lounge 109