Uniquely $D$-Colourable Digraphs with Large Girth II: Simplification via Generalization

P Mark Kayll,Esmaeil Parsa

doi:10.37236/9689

Abstract

We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $\psi\colon D^*\to D$ such that: (i) for every digraph $C$ of order at most $k$, there exists an acyclic homomorphism $D^*\to C$ if and only if there exists an acyclic homomorphism $D\to C$; and (ii) for every $D$-pointed digraph $C$ of order at most $k$ and every acyclic homomorphism $\varphi\colon D^*\to C$ there exists a unique acyclic homomorphism $f\colon D\to C$ such that $\varphi=f\circ\psi$. This implies the main results in [A. Harutyunyan et al., Uniquely $D$-colourable digraphs with large girth, Canad. J. Math., 64(6) (2012), 1310—1328; MR2994666] analogously with how the work [J. Nešetřil and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B, 90(1) (2004), 161—172; MR2041324] generalizes and extends [X. Zhu, Uniquely $H$-colorable graphs with large girth, J. Graph Theory, 23(1) (1996), 33—41; MR1402136].

Highlights

One authoritative combinatorialist went so far as to assert that “All interesting combinatorics flows from the existence of graphs with large girth and chromatic number.”1 we interpret Thomasse’s remark as somewhat tongue-in-cheek, but as they say, many a truth is said in jest
We show that the event that Wi1 ∪ Wi2 ∪ · · · ∪ Wis induces an acyclic subdigraph in D∗ is unlikely
By Markov’s Inequality, the probability that there exists such a set {z} ∪ U that induces an acyclic subdigraph is less than e−n1+ /2, which means it is unlikely as desired

Summary

Introduction

In 1959, Paul Erdos, in a landmark paper [7]— known as one of the most pleasing uses of the probabilistic method—proved the existence of graphs with arbitrarily large girth. Because our Theorem 1 likewise characterizes when the high directed girth, high digraph chromatic number (for unique colourability) phenomenon occurs—phrased in terms of acyclic homomorphisms—it too reaches a satisfying destination, for the sequence [4, 9]. Because this level of generality has shortened the proofs from [9], perhaps we’ve arrived at the ‘right’ vantage point for viewing these results.

Set-up for the proof of Theorem 1

Proof of Theorem 1