# Graphs & Digraphs with Large Chromatic Numbers & Long Shortest Cycles

## Document Type

Presentation Abstract

## Presentation Date

9-26-2002

## Abstract

To get an idea of how extensive is the chromatic theory of graphs, one can open any graph theory book and read the chapters on coloring. To get an idea of how limited is the same theory, one can simply open a particular book, *Graph Coloring Problems*, a 300-page monograph listing primarily open problems in the subject. Thus we observe a dichotomy: graph coloring enjoys a rich, yet largely incomplete, theory. Since the problems are out there and the foundations are strong, interesting new theorems appear virtually every year.

In this colloquium, I'll introduce some basic graph coloring notions and explore a famous paradoxical theorem of Paul Erdös from 1959. This will set the stage for a natural extension of the chromatic number concept to directed graphs, also known as *digraphs*. Finally, I'll present a digraph analogue of Erdös' theorem. The proof, discovered in Slovenia last November, uses probabilistic ideas and a surprising application of a fact from basic algebra. *Which* fact shall remain top secret until the key moment in the talk when it is needed.

## Recommended Citation

Kayll, Dr. Mark, "Graphs & Digraphs with Large Chromatic Numbers & Long Shortest Cycles" (2002). *Colloquia of the Department of Mathematical Sciences*. 126.

https://scholarworks.umt.edu/mathcolloquia/126

## Additional Details

Thursday, 26 September 2002

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)