“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.

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

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