An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any
non-identity automorphism of the graph. The motion of a graph is the minimal degree of its …

In the past, analogies to Brooks' theorem have been found for various parameters of graph
coloring for infinite locally finite connected graphs in ZFC. We prove these theorems are not …

A graph is called uniquely distinguishing colorable if there is only one partition of vertices of
the graph that forms distinguishing coloring with the smallest possible colors. In this paper …

We prove that Brook's type theorems for several coloring parameters of infinite graphs,
which are true in ZFC, are not provable in ZF (ie, the Zermelo–Fraenkel set theory without …

Abstract Let G⩽ Sym (X) G\leqslantSym(X) be a group of permutations of a countable set X
X. Call a colouring of XX asymmetric if no g∈ G∖ id g∈G∖{id} preserves all colours. The …

Let $ X $ be a (repetitive) infinite connected simple graph with a finite upper bound $\Delta $
on the vertex degrees. The main theorem states that $ X $ admits a (repetitive) limit …

Call a colouring of a graph distinguishing if the only automorphism which preserves it is the
identity. We investigate the role of the Axiom of Choice in the existence of certain proper or …

A vertex colouring of a graph is asymmetric if it is preserved only by the identity
automorphism. The minimum number of colours needed for an asymmetric colouring of a …

The distinguishing number or asymmetric coloring number of a group G on the set Ω,
denoted D (G, Ω) or ACN (G, Ω) is the smallest k such that there is a coloring of Ω by k colors …

Call a vertex or an edge colouring of a graph distinguishing, if is not preserved by any non-
identity automorphism. For a graph H, we say that a graph G is H-free if there is no induced …