Distinguishing homomorphisms of infinite graphs

TL;DR

This paper establishes bounds on the distinguishing chromatic number of certain infinite graphs and introduces the concept of distinguishing homomorphisms, showing their abundance under specific conditions.

Contribution

It introduces the notion of distinguishing homomorphisms for infinite graphs and proves their existence in graphs with the connected existentially closed property.

Findings

Infinite graphs with certain properties admit continuum-many distinguishing homomorphisms.

Provides bounds on the distinguishing chromatic number for specific infinite graphs.

Applications to universal H-colorable graphs with finite cores.

Abstract

We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper $n$-colourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph $G$ satisfies the connected existentially closed property and admits a homomorphism to $H$, then it admits continuum-many distinguishing homomorphisms from $G$ to $H$ join $K_2.$ Applications are given to a family universal $H$-colourable graphs, for $H$ a finite core.