On random digraphs and cores

TL;DR

This paper proves that for certain probabilities, random digraphs are almost surely cores, extending known results from random graphs to directed graphs.

Contribution

It establishes that under specific conditions, random digraphs are asymptotically almost surely cores, generalizing prior results from undirected to directed graphs.

Findings

Random digraphs are almost surely cores for certain p(n).

Extension of random graph core results to directed graphs.

Provides probabilistic thresholds for core properties in digraphs.

Abstract

An acyclic homomorphism of a digraph $C$ to a digraph $D$ is a function $\rho\colon V(C)\to V(D)$ such that for every arc $uv$ of $C$, either $\rho(u)=\rho(v)$, or $\rho(u)\rho(v)$ is an arc of $D$ and for every vertex $v\in V(D)$, the subdigraph of $C$ induced by $\rho^{-1}(v)$ is acyclic. A digraph $D$ is a core if the only acyclic homomorphisms of $D$ to itself are automorphisms. In this paper, we prove that for certain choices of $p(n)$, random digraphs $D\in D(n,p(n))$ are asymptotically almost surely cores. For digraphs, this mirrors a result from [A. Bonato and P. Pra{\l}at, The good, the bad, and the great: homomorphisms and cores of random graphs, Discrete Math., 309 (2009), no. 18, 5535-5539; MR2567955] concerning random graphs and cores.