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Perfect Zero-Divisor Graphs of Z N Rose-Hulman Undergraduate Mathematics Journal Volume 17 Issue 2 Article 6 Perfect Zero-Divisor Graphs of Z_n Bennett Smith Illinois College Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Smith, Bennett (2016) "Perfect Zero-Divisor Graphs of Z_n," Rose-Hulman Undergraduate Mathematics Journal: Vol. 17 : Iss. 2 , Article 6. Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss2/6 Rose- Hulman Undergraduate Mathematics Journal Perfect Zero-Divisor Graphs of Zn Bennett Smitha Volume 17, No. 2, Fall 2016 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 [email protected] a scholar.rose-hulman.edu/rhumj Illinois College Rose-Hulman Undergraduate Mathematics Journal Volume 17, No. 2, Fall 2016 Perfect Zero-Divisor Graphs of Zn Bennett Smith Abstract. The zero-divisor graph of a ring R, denoted Γ(R), is the graph whose vertex set is the collection of zero-divisors in R, with edges between two distinct vertices u and v if and only if uv = 0. In this paper, we restrict our attention to Γ(Zn), the zero-divisor graph of the ring of integers modulo n. Specifically, we determine all values of n for which Γ(Zn) is perfect. Our classification depends on the prime factorization of n, with relatively simple prime factorizations corresponding to perfect graphs. In fact, for values of n with at most two distinct prime factors, Γ(Zn) is perfect; for n with at least 5 distinct prime factors, Γ(Zn) is not perfect; and for n with either three or four distinct prime factors, Γ(Zn) is perfect only in the cases where n = paqr and n = pqrs for distinct primes p; q; r; s and positive integer a. In proving our results, we make heavy use of the Strong Perfect Graph Theorem. Page 114 RHIT Undergrad. Math. J., Vol. 17, No. 2 1 Introduction Consider the set of zero-divisors of Zn, denoted ZD(Zn). The zero-divisor graph of Zn, denoted Γ(Zn), is the graph whose vertex set is ZD(Zn), where distinct vertices u and v are adjacent if and only if uv ≡n 0. For example, Figure 1.1 illustrates the zero-divisor graph of Z12. Zero-divisor graphs can be defined more generally for any ring R, but the focus of this paper is on zero-divisor graphs of Zn. Figure 1.1: The zero-divisor graph of Z12. Zero-divisor graphs were first introduced in 1988 by Beck, where the primary focus was on the coloring of the graphs [4]. Several decades later, the concept was further explored by Anderson and Livingston, whose work connecting the areas of commutative ring theory and graph theory continues to encourage further research on zero-divisor graphs [2]. Beck's focus is applicable to more recent research done on proper colorings of zero-divisor graphs, such as by Duane [6]. The zero-divisor graph survey paper by Anderson, Axtell, and Stickles summarizes much of this research [1]. Endean, Henry, and Manlove attempted to determine all values of n for which the zero- divisor graph of Zn is perfect [7]. Although they deserve much credit for introducing this question, their work contains mistaken results. Here, we correctly determine all values of n for which Γ(Zn) is perfect. To do so, we make use of the Strong Perfect Graph Theorem, a beautifully simple characterization of perfect graphs, proved by Chudnovsky, Robertson, Seymore, and Thomas [5]. We also use a simplification of the zero-divisor graph of Zn, which we refer to as the zero-divisor type graph of Zn. Recently, we have learned that this concept has been discovered before and used in earlier works. For example, these simplified graphs are presented more generally and referred to as compressed graphs by Axtell, Baeth, and Stickles [3] and by Weber [9]. Additionally, these simplified graphs are referred to as graphs of equivalency classes of zero-divisors in Zn by Spiroff and Wickham [8]. As others before us, we have found this simplified view of zero-divisor graphs to be very useful. In Section 2, we discuss the definition of a perfect graph. In Section 3, we discuss the simplification of zero-divisor graphs to zero-divisor type graphs. Then in Section 4, we prove that the zero-divisor graph is perfect if and only if the zero-divisor type graph is perfect. Using this theorem and the Strong Perfect Graph Theorem, in Sections 5 and 6 we completely determine all values of n for which the zero-divisor graph of Zn is perfect, leading to our main theorem. RHIT Undergrad. Math. J., Vol. 17, No. 2 Page 115 Main Theorem. The zero-divisor graph of Zn, Γ(Zn), is perfect if and only if n has one of the following four forms. 1. n = pa for a prime p and positive integer a 2. n = paqb for distinct primes p; q and positive integers a; b 3. n = paqr for distinct primes p; q; r and positive integer a 4. n = pqrs for distinct primes p; q; r; s: 2 Perfectness of a Graph In this section, we define what it means for a graph to be perfect and introduce a simple way to determine the perfectness of any graph. The definition of a perfect graph relies upon the following two terms. The chromatic number of a graph G is the minimum number of colors required to color the vertices of G such that no two adjacent vertices have the same color. The clique number of graph G is the size of the largest complete subgraph of G. This leads to the following definition. Definition. A perfect graph is a graph G for which every induced subgraph of G has chromatic number equal to its clique number. Example 2.1. Consider the graph of a 5-cycle, shown in Figure 2.1. The chro- matic number of the graph is 3, while the clique number is 2. Thus, the graph of a 5-cycle is not perfect. Figure 2.1: The most efficient coloring of a 5-cycle. The number of colors required is three, so the chromatic number is 3. Now, consider the graph of a 4-cycle, shown in Figure 2.2. Here, the chromatic number and the clique number of the graph are 2. Thus, the chromatic number and the clique number are equal. For the graph to be perfect the clique number and chromatic number must be equal for every induced subgraph as well. This is easily verified, and hence the graph of a 4-cycle is perfect. Page 116 RHIT Undergrad. Math. J., Vol. 17, No. 2 Figure 2.2: The most efficient coloring of a 4-cycle. The number of colors required is two, so the chromatic number is 2. Another way to check the perfectness of these two graphs is with the Strong Perfect Graph Theorem, which reaffirms that no cycle of odd length at least 5 is perfect, while every cycle of even length at least 4 is perfect [5]. First, note that a hole of a graph G is an induced subgraph in the form of a cycle of length at least 4; whereas an antihole is an induced subgraph of G whose complement is a hole in G. Also, note that the length of a hole or antihole refers to the number of vertices it contains. Strong Perfect Graph Theorem. A graph G is perfect if and only if it contains no hole or antihole of odd length. 3 Zero-Divisor Type Graphs In this section, we define a graph called the zero-divisor type graph of Zn, denoted T Γ (Zn). We note that the zero-divisor type graph is identical to the graph of equivalency classes of zero-divisors in Zn. Graphs of equivalency classes of zero-divisors in a ring R were first defined by Spiroff and Wickham [8]. Other authors have referred to these graphs as compressed zero-divisor graphs [3, 9]. Here, we discuss the same concept in the case were R = Zn and we refer to these graphs as zero-divisor type graphs. We will see that the zero-divisor type graph is very closely related to the zero-divisor graph. However, the zero-divisor type graph is much simper in most cases, while maintaining the property of perfectness (proven in Section 4). We begin by defining the vertex and edge sets of zero-divisor type graphs. Note that a nontrivial divisor of n is a divisor of n that is neither 1 nor n. T T Vertices of Γ (Zn). The set of vertices of Γ (Zn) is given by fTd : d is a nontrivial divisor of ng; where Td is the subset of ZD(Zn) defined by Td = fk 2 ZD(Zn) : gcd(k; n) = dg: T T Edges of Γ (Zn). Two distinct vertices Td1 and Td2 are connected by an edge in Γ (Zn) if and only if d1d2 ≡n 0. RHIT Undergrad. Math. J., Vol. 17, No. 2 Page 117 T We refer to each vertex of Γ (Zn) as a zero-divisor type, or simply a type. We say that a zero-divisor k is of type d or in type Td to mean that k 2 Td. We use the terms zero-divisor type graph and type graph to refer to the same thing. Before making a few key observations about the type graph of Zn, we present an example. Example 3.1. Consider Z12. The set of zero-divisors in Z12 is ZD(Z12) = f2; 3; 4; 6; 8; 9; 10g; and the zero-divisor graph, Γ(Z12), can be seen in Figure 3.1.
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