# “D-colorable digraphs with large girth”

## Document Type

Presentation Abstract

## Presentation Date

5-11-2011

## Abstract

In 1959 Paul Erds (Canad. J. Math. **11** (1959), 34-38) famously proved, nonconstructively, that there exist graphs that have both arbitrarily large girth and arbitrarily large chromatic number. This result, along with its proof, has had a number of descendants that have extended and generalized the result while strengthening the techniques used to achieve it. We follow the lead of Xuding Zhu (J. Graph Theory **23** (1996), 33-41) who proved that, for a suitable graph *H*, there exist graphs of arbitrarily large girth that are uniquely *H*-colorable (a homomorphism property generalizing coloring). We establish an analogue of Zhu's result in a digraph setting with a certain type of homomorphism.

Let *C* and *D* be digraphs. A mapping ƒ : *V* (*D*) → *V* (*C*) is a C-coloring if for every arc *uv* of *D*, either ƒ(*u*)ƒ(*v*) is an arc of *C* or ƒ(*u*) = ƒ(*v*), and the preimage of every vertex of *C* induces an acyclic subdigraph in D. We say that *D* is C-colorable if it admits a *C*-coloring and that *D* is uniquely *C*-colorable if it is surjectively *C*-colorable and any two *C*-colorings of *D* differ by an automorphism of *C*. In this dissertation, we prove that if *D* is a digraph that is not *C*-colorable, then there exist graphs of arbitrarily large girth that are *D*-colorable but not *C*-colorable. Moreover, for every digraph *D* that is uniquely *D*-colorable, there exists a uniquely *D*-colorable digraph of arbitrarily large girth. In this talk, we sketch the proof of the former result, taking care to stress the main techniques over the fine details.

## Recommended Citation

Rafferty, Liam, "“D-colorable digraphs with large girth”" (2011). *Colloquia of the Department of Mathematical Sciences*. 376.

https://scholarworks.umt.edu/mathcolloquia/376

## Additional Details

Doctoral Dissertation Defense. Link to the presenter's dissertation.

Dissertation Committee:P. Mark Kayll, Chair (Mathematical Sciences), Min Chen (Computer Science),

Solomon Harrar (Mathematical Sciences), Jennifer McNulty (Mathematical Sciences), George McRae (Mathematical Sciences) Wednesday, May 11, 2011

1:10 pm in Math 103