## Graduate Student Theses, Dissertations, & Professional Papers

#### Title

ASPECTS OF UNIQUE D-COLORABILITY FOR DIGRAPHS

2019

#### Document Type

Dissertation - Campus Access Only

#### Degree Type

Doctor of Philosophy (PhD)

Mathematics

#### Department or School/College

Department of Mathematical Sciences

P. Mark Kayll

#### Commitee Members

Eric Chesebro, Kelly McKinnie, Cory Palmer, Travis Wheeler

#### Keywords

Acyclic chromatic Number, Acyclic coloring, Acyclic homomorphisms, Digraph girth, High girth high chromatic number

#### Publisher

University of Montana

#### Abstract

We prove that for every digraph $C$ and every choice of positive integers $k$ and $\ell$ there exists a digraph $D$ with girth at least $\ell$ together with a surjective acyclic homomorphism $c:D\rightarrow C$ such that (i) for every digraph $C'$ with at most $k$ vertices, there exists an acyclic homomorphism $g:D\rightarrow C'$ if and only if there exists an acyclic homomorphism $f:C\rightarrow C'$ and (ii) for every $C$-pointed digraph $C'$ with at most $k$ vertices and for every acyclic homomorphism $g:D\rightarrow C'$ there exists a unique acyclic homomorphism $\phi:C\rightarrow C'$ such that $g=\phi\circ c$. This implies the main results in [A. Harutyunyan et al., Uniquely $D$--colourable digraphs with large girth, \textit{Canad.\ J. Math.}, \textbf{64(6)} (2012), 1310--1328]. We also show that the two definitions of uniquely $D$-colorable digraphs that are either in terms of automorphisms or by vertex partitions are not always equivalent and study conditions under which they are equivalent. In response to the question for what portion of digraphs do the aforementioned conditions hold, using the probabilistic method, we prove that asymptotically almost surely every random digraph is a core for which these conditions do not hold.

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