# Digraphs and Homomorphisms: Cores, Colorings, and Constructions

## Document Type

Presentation Abstract

## Presentation Date

5-14-2014

## Abstract

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an acyclic set, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets. In the spirit of a famous theorem of P. Erdős [Graph theory and probability, Canad. J. Math. 11:34–38, (1959)], it was shown probabilistically in [D. Bokal et al., The circular chromatic number of a digraph, J. Graph Theory **46**(3): 227–240, (2004)] that there exist digraphs with arbitrarily large digirth and chromatic number. Here we give a construction of such digraphs and define a new product of these highly chromatic digraphs with the directed analogue of the complete graph. This product gives a construction of uniquely *n*-colorable digraphs without short cycles.

The graph-theoretic notion of ‘homomorphism’ also gives rise to a digraph analogue. An acyclic homomorphism from a digraph *D* to a digraph *H* is a mapping 𝜑 : *V *(*D*) → *V* (*H*) such that *uv* ∈ *A* (*D*) implies that either 𝜑(*u*)𝜑(*v*) ∈ *A*(*H*) or 𝜑(*u*) = 𝜑(*v*), and all the 'fibers' 𝜑^{-1}(*v*), for *v* ∈ *V*(*H*), of 𝜑 are acyclic. In this language, a core is a digraph *D* for which there does not exist an acyclic homomorphism from *D* to itself. Here we prove some basic results about digraph cores and construct new classes of cores. We also define a digraph-theoretic analogue to the graph-theoretic ‘fractional chromatic number’ and prove results relating it to other well-known digraph invariants. We see that it behaves similarly to the graph-theoretic analogue.

## Recommended Citation

Severino, Michael, "Digraphs and Homomorphisms: Cores, Colorings, and Constructions" (2014). *Colloquia of the Department of Mathematical Sciences*. 452.

https://scholarworks.umt.edu/mathcolloquia/452

## Additional Details

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

Dissertation Committee: Dr. Mark Kayll, Chair (Mathematical Sciences), Dr. Kelly McKinnie (Mathematical Sciences), Dr. Jenny McNulty (Mathematical Sciences), Dr. George McRae (Mathematical Sciences), Dr. Mike Rosulek (Computer Science – Oregon State University)Wednesday, May 14, 2014 at 9:10 am in Math 312