Journal of Mathematics

Volume 2017, Article ID 7458318, 7 pages

https://doi.org/10.1155/2017/7458318

## Chromatic Numbers of Suborbital Graphs for the Modular Group and the Extended Modular Group

Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand

Correspondence should be addressed to Pradthana Jaipong; ht.ca.umc@j.anahtdarp

Received 20 October 2016; Revised 19 January 2017; Accepted 20 February 2017; Published 5 March 2017

Academic Editor: Francisco B. Gallego

Copyright © 2017 Wanchai Tapanyo and Pradthana Jaipong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This research studies the chromatic numbers of the suborbital graphs for the modular group and the extended modular group. We verify that the chromatic numbers of the graphs are or . The forest conditions of the graphs for the extended modular group are also described in this paper.

#### 1. Introduction

*Graph coloring* is a special case of graph labeling which assigns labels to all vertices or edges of a given graph. In this special case, the labels are called* colors*.* Vertex coloring* is the study of how many colors can be assigned to the vertices of a graph verifying that no adjacent vertices are equally colored. Similarly,* edge coloring* assigns colors to edges of a graph that the same color cannot be painted to edges which have a common vertex.

The graph which can be assigned colors on its vertices is said to be *-colorable*. If is the smallest number of colors, the graph is said to be *-chromatic* and is called the* chromatic number* of the graph. In the case of edges, we add the word “edge” to the terminologies, for example, *-edge-colorable*, *-edge-chromatic*, and* edge chromatic number*.

This paper studies coloring of suborbital graphs for the modular group and the extended modular group. The results of edge coloring are trivial, so we focus on vertex coloring. This study is separated into two parts in Sections 3 and 4. In Section 3, we provide the coloring results of the directed graph , the suborbital graph for the modular group . The graph was first constructed in [1] using the general concept introduced by Sims; see more in [2]. The authors showed that was the disjoint union of isomorphic copies of their special subgraph , the* generalized Farey graph*, coming from the use of the subgroup of . Then many properties of were investigated through the studies on . Certainly, vertex coloring of is also examined through the study on .

In 2013, the concept of suborbital graphs continued to the extended modular group , as in [3]. The notations are defined likewise. That is, denotes the suborbital graph for the extended modular group , and is a special subgraph of coming from the subgroup of . Certainly, is the disjoint union of isomorphic copies of . The vertex coloring results of this case are described in Section 4. The forest conditions for the graph , which are necessary in vertex coloring, are also studied in this section.

Note that the graph in the context of coloring is loopless. Thus, coloring of the suborbital graphs with loops will not be mentioned.

#### 2. Preliminaries

In this section, we provide some necessary backgrounds used in coloring. Some backgrounds of suborbital graphs may not be mentioned in this research. We start this section with the definition of the modular group. The modular group is the projective special linear group , the group of matrices of unit determinant with integer coefficients such that every matrix is identified with its negative . Therefore, is the quotient of the special linear group by its center , where is the identity matrix of order . In other words, the elements of are represented by the pairs of matrices , where and . We usually leave out the signs of the representations for convenience.

From another point of view, the group may be presented as a group of transformations. In this manner, it is the group of linear fractional transformations on where the composition of functions is the group operation. Certainly, the group in this manner and are isomorphic. In other words, is considered as a group acting on by linear fractional transformation, that is,where and . This action can be also extended to the set as follows: where is an irreducible fraction representing an element of . In this case, is represented by and . Certainly, the fractions representing the elements of are not unique since . However, it does not affect the action of on . Notice that the action of on is transitive; that is, has only one orbit under the action of .

The extended modular group is a group generated by and the hyperbolic reflection . In the forms of matrix representations, , that is,On the set , the complex conjugate is ineffective. Then the action of on is the same as that of the group .

In [1], a suborbital graph for on was established. The vertex set is defined to be the set . The set of directed edges depends on a pair . It is defined to be the orbit where the action of on the Cartesian is the component-wise action extended from that on . In this research, we let denote a directed edge from to ; that is, there exists the edge of the graph if . Since acts on transitively, there is an element such that . Certainly, , so it can be represented as a reduced fraction with nonnegative denominator. This implies that . Hence . Then the suborbital graph is simply denoted by . The case is the case of trivial suborbital graphs. There is one loop at every vertex. The coloring of this case will not be mentioned. In the case , edges of the graph are defined to be the upper half-circles and the vertical half-lines in joining elements of ; see Figure 1 for examples. In this case, the graph is determined. It is a restriction of on the orbit , where is a congruence subgroup of defined by Note that . In fact, is a suborbital graph for on . The following remark is the result obtained directly after constructing the -invariant equivalence relation related to ; see more details in [1].