A homomorphism from one graph to another is a map that sends vertices to vertices and edges to edges. We denote the number of homomorphisms from G to H by |G → H|. If F is a collection of graphs, we say that F distinguishes graphs G and H if there is some member X of F such that |G → X| 6= |H → X|. F is a distinguishing family if it distinguishes all pairs of graphs. We show that various collec...

Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X, denoted DG(X), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we show that if G is nilpotent of class c or supersolvable of length c then G always acts with distinguishing number a...

The distinguishing chromatic number χD(G) of a graph G is the minimum number of colours required to properly colour the vertices of G so that the only automorphism of G that preserves colours is the identity. For a graph G of order n, it is clear that 1 6 χD(G) 6 n, and it has been shown that χD(G) = n if and only if G is a complete multipartite graph. This paper characterizes the graphs G of o...

Collins and Trenk define the distinguishing chromatic number χD(G) of a graph G to be the minimum number of colors needed to properly color the vertices of G so that the only automorphism of G that preserves colors is the identity. They prove results about χD(G) based on the underlying graphG. In this paper we prove results that relate χD(G) to the automorphism group of G. We prove two upper bo...

Balogh, Bollobás and Weinreich showed that a parameter that has since been termed the distinguishing number can be used to identify a jump in the possible speeds of hereditary classes of graphs at the sequence of Bell numbers. We prove that every hereditary class that lies above the Bell numbers and has finite distinguishing number contains a boundary class for well-quasi-ordering. This means t...

Introduced by Albertson and al. [1], the distinguishing number D(G) of a graph G is the least integer r such that there is a r-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs not depend on n. In this pa...