"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, XD(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 XD: if G is a connected graph with maximum degree Δ, then XD(G) ≤ 2Δ, with equality if and only if G is KΔ,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.

Additional Details

Wednesday, 19 May 2010
2:10 p.m. in Math 103
3:00 p.m. Refreshments in Math Lounge 109

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