# Homomorphisms and cores of random digraphs

## Presentation Type

Oral Presentation

## Abstract/Artist Statement

An acyclic coloring of the vertices of a digraph D is an assignment of different colors to the vertices of D such that the vertices of the same color induce an acyclic subdigraph of D. If an acyclic coloring of D uses k distinct colors, it is called an acyclic k-coloring of D. The minimim number k for which there is an acyclic k-coloring of D is called the acyclic chromatic number of D. It is easy to see that the acyclic chromatic number od D is equal to k if and only if we can define a mapping f from V(D) to V(K_k) such that f maps an arc either to a vertex or to an arc of K_k with direction preserved and the inverse image of every vertex of K_k induces an acyclic subdigraph of D. A mapping f:V(D)---> V(C) with the forementioned properties is called an acyclic homomorphism of D to C. If there is an acyclic homomorphism of D to C, we say D is C-colorable. A one-one and onto homomorphism is called an isomorphism and an isomorphism of D to itself is called an automorphism. A digraph D is called a core if the only homomorphism of D to itself is an automorphism. Complete digraphs are examples of cores.

Cores play a very important role in the study of digraph homomorphisms. A very first question about cores is that ``how big is the set of core digraphs?''. In this talk we want to prove that assymptotically almost surely every random digraph is a core. Loosely speaking this means that if we have a bag containing all digraphs on n vertices and we randomly choose a digraph from the bag, the probability that the outcome is a core digraph tends to 1 as n goes to infinity.

## Mentor Name

Mark Kayll

Homomorphisms and cores of random digraphs

UC 333

An acyclic coloring of the vertices of a digraph D is an assignment of different colors to the vertices of D such that the vertices of the same color induce an acyclic subdigraph of D. If an acyclic coloring of D uses k distinct colors, it is called an acyclic k-coloring of D. The minimim number k for which there is an acyclic k-coloring of D is called the acyclic chromatic number of D. It is easy to see that the acyclic chromatic number od D is equal to k if and only if we can define a mapping f from V(D) to V(K_k) such that f maps an arc either to a vertex or to an arc of K_k with direction preserved and the inverse image of every vertex of K_k induces an acyclic subdigraph of D. A mapping f:V(D)---> V(C) with the forementioned properties is called an acyclic homomorphism of D to C. If there is an acyclic homomorphism of D to C, we say D is C-colorable. A one-one and onto homomorphism is called an isomorphism and an isomorphism of D to itself is called an automorphism. A digraph D is called a core if the only homomorphism of D to itself is an automorphism. Complete digraphs are examples of cores.

Cores play a very important role in the study of digraph homomorphisms. A very first question about cores is that ``how big is the set of core digraphs?''. In this talk we want to prove that assymptotically almost surely every random digraph is a core. Loosely speaking this means that if we have a bag containing all digraphs on n vertices and we randomly choose a digraph from the bag, the probability that the outcome is a core digraph tends to 1 as n goes to infinity.