Number of distinguishing colorings and partitions
Introduction
Breaking symmetries of graphs via vertex coloring is a subject initiated by Babai’s work [3] in 1977. There, he introduced an asymmetric coloring of a graph, and proved that a tree has an asymmetric coloring with two colors if all vertices have the same degree , where can be an arbitrary finite or infinite cardinal. The concept was later called distinguishing coloring in the literature, since the appearance of [1] by Albertson and Collins in 1996.
This paper is about counting non-equivalent distinguishing colorings and partitions of a given graph with colors. Considering this number in case of 2 colors is as old as symmetry breaking in graphs; in 1977 Babai [3] tried to count distinguishing colorings of infinite trees, while in 1991 Polat and Sabidussi [19] tried to count essentially different asymmetrising sets in finite and infinite trees which are distinguishing (coloring) partitions with cells in our terminology (see Section 4).
A vertex coloring of a graph is called distinguishing if it is only preserved by the identity automorphism; in this case, we say that the coloring breaks all the symmetries of . By a -distinguishing coloring we mean a distinguishing coloring which uses exactly colors. The distinguishing number of a graph , denoted , is the smallest number such that there exists a distinguishing vertex coloring of with colors. A graph is called -distinguishable if there exists a distinguishing vertex coloring with colors [1]. The distinguishing number of some important classes of graphs are as follows: , , for , while for [1].
A whole wealth of results on the subject has already been generated. Among many, we can only mention a few, only those that have essentially important results, or those that have introduced new indices based on distinguishing colorings. For a connected finite graph , it was independently proved by Collins and Trenk [5] and Klavžar et al. [18] that , where is the largest degree of . Equality holds if and only if is a complete graph , a balanced complete bipartite graph , or . Collins and Trenk [5] also mixed the concept of distinguishing colorings with proper vertex colorings to introduce the distinguishing chromatic number of a graph . It is defined as the minimum number of colors required to properly color the vertices of such that this coloring is only preserved by the trivial automorphism. They also showed that, for a finite connected graph , we have and that equality holds only if is isomorphic to or .
Symmetry breaking can also happen via other kinds of graph colorings. An analogous index for an edge coloring, namely the distinguishing index , has been introduced by Kalinowski and Pilśniak in [16] as the minimum number of the required colors in an asymmetric edge-coloring of a connected graph . Moreover, they showed that for a finite connected graph , unless is isomorphic to , or . Another analogous index is introduced by Kalinowski, Pilśniak and Woźniak in [17]; the total distinguishing number is the minimum number of required colors in an asymmetric total coloring of .
To generalize some results from the finite case to the infinite ones, Imrich, Klavžar and Trofimov [14] considered the distinguishing number for infinite graphs. They showed that for an infinite connected graph we have , where is a cardinal number such that the degree of any vertex of is not greater than . Most symmetry breaking concepts have their relative counterparts in the infinite case, however, there are some (such as the Infinite Motion Conjecture) that arise only when we consider infinite graphs. As an instance, one can take a look at [13] by Imrich et al. which contains comparisons of some distinguishing indices of connected infinite graphs.
It was also interesting to know the distinguishing number for product graphs. For example, Bogstad and Cowen [4] showed that for , every hypercube of dimension , which is the Cartesian product of copies of , is -distinguishable. It has also been shown by Imrich and Klavžar in [15] that the distinguishing number of Cartesian powers of a connected graph is equal to two except for . Meanwhile, Imrich, Jerebic and Klavžar [12] showed that Cartesian products of relatively prime graphs whose sizes are close to each other can be distinguished with a small number of colors. Moreover, Estaji et al. in [8] proved that for every pair of connected graphs and with , we have . Gorzkowska, Kalinowski and Pilśniak proved a similar result for the distinguishing index of the Cartesian product [10].
The lexicographic product was a subject of symmetry breaking via vertex and edge coloring by Alikhani and Soltani in [2], where they showed that under some conditions on the automorphism group of a graph , we have , where is the th lexicographic power of , for any natural number . As well, they showed that if and are connected graphs, then .
Coloring is not the only mean of symmetry breaking in graphs. For example, one might break the symmetries of a graph via a more general tool such as vertex partitioning. Ellingham and Schroeder introduced distinguishing partition of a graph as a partition of the vertex set that is preserved by no nontrivial automorphism [7]. Here, unlike coloring, some graphs have no distinguishing partition. Anyhow, for a graph that admits a distinguishing partition, one may think of the minimum number of required cells in a distinguishing partition of the vertex set. Here, we show this index by .
In this paper, we introduce some further indices related to symmetry breaking of graphs by studying the number of non-equivalent distinguishing colorings of a graph with colors and some other similar quantities. This is motivated by the problem of evaluating the distinguishing number of a lexicographic product or the -join of some graphs, which we consider in Section 5.
The paper is organized as follows. In Section 2, we consider the number of non-equivalent distinguishing colorings of a graph with (exactly) colors, namely (and ) and, we calculate these indices for some simple types of graphs. Afterwards in Section 3, we introduce the distinguishing threshold as a dual index to the distinguishing number. It is shown that calculations of some indices introduced in Sections 2 Non-equivalent distinguishing colorings, 4 Non-equivalent distinguishing partitions are easier, in some cases, when we know the distinguishing threshold. Moreover, in Section 4, we consider the number of non-equivalent distinguishing coloring partitions of a graph with (exactly) cells, namely (and ) and, we calculate them in some special cases. Additionally in Section 4, some other auxiliary indices are also introduced. Then, we present an application of one of the indices introduced here, namely , in Section 5. We finally conclude the paper by shedding some lights on the future investigations in Section 6.
Here, we use the standard notation and terminology of graph theory, which can be found in [6]. We only remind that the set of neighbors of a vertex in G is denoted by , while stands for the set .
Section snippets
Non-equivalent distinguishing colorings
Two colorings and of a graph are called equivalent if there is an automorphism of such that for all .
The number of non-equivalent distinguishing colorings of a graph with as the set of admissible colors is shown by , while the number of non-equivalent -distinguishing colorings of a graph with as the set of colors is shown by . When has no distinguishing colorings with exactly colors, we put . It is also clear that
Distinguishing threshold
For any graph , we define the distinguishing threshold to be the minimum number such that for any , any arbitrary coloring of with colors is distinguishing. For example and . Note, also, that for an asymmetric graph , we have . Moreover, we always have .
Let be a graph on vertices. If any two distinct vertices have different set of neighbors other than themselves, then any -coloring of has to be distinguishing because
Non-equivalent distinguishing partitions
In this section, we turn our attention to the case of different distinguishing partitions of graphs. Let be a graph and let and be two partitions of the vertices of . We say and are equivalent if there is a non-trivial automorphism of which maps onto . The number of non-equivalent partitions of , with at most cells, is called the partition number of and is denoted by .
Meanwhile, a distinguishing coloring partition of a graph is a partition of the vertices of
Distinguishing lexicographic products
In this section we provide an important application of one of the indices introduced in this paper, namely . We start by recalling some preliminaries to the topic of lexicographic product of graphs.
Let be a graph. The -join of , is the graph with and Whenever, for all , we have , for a fixed graph , the graph is called the lexicographic product of and and we write .
We remind the
Conclusion
We have seen in Section 5 that counting the number of non-equivalent distinguishing colorings, , has an application in finding the distinguishing number of lexicographic products. Moreover, other indices have shown to have deep interactions with each other and also with . It should be noted that calculating these indices are not always easy. Even when the automorphism group is very small and simple, counting non-equivalent distinguishing colorings or distinguishing (coloring) partitions faces
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
We owe a great debt to professor Wilfried Imrich, who proposed several problems, which led to this paper, during his visit to Shiraz University.
References (20)
- et al.
The distinguishing number of hypercubes
Discrete Math.
(2004) The group of an X-join of graphs
J. Combin. Theory
(1968)- et al.
The distinguishing number of Cartesian products of complete graphs
European J. Combin.
(2008) - et al.
Distinguishing graphs by edge-colourings
European J. Combin.
(2015) - et al.
Distinguishing labelings of group action on vector spaces and graphs
J. Algebra
(2006) - et al.
Asymmetrising sets in trees
Discrete Math.
(1991) - et al.
Symmetry breaking in graphs
Electron. J. Combin.
(1996) - et al.
The distinguishing number and distinguishing index of the lexicographic product of two graphs
Discuss. Math. Graph Theory
(2018) Asymmetric trees with two prescribed degrees
Acta Math. Acad. Sci. Hung.
(1977)- et al.
The distinguishing chromatic number
Electron. J. Combin.
(2006)