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Fractional Colorings and the Mycielski Graphs

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Fractional Colorings and the Mycielski Graphs FRACTIONAL COLORINGS AND THE MYCIELSKI GRAPHS By G. Tony Jacobs Under the Direction of Dr. John S. Caughman A math 501 project submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics at the PORTLAND STATE UNIVERSITY Fall 2006 i List of Figures 1 Cycle and complete graph on ¯ve vertices . 3 2 The graphs C5 and K5 with proper 3- and 5-colorings, respectively . 5 3 The graph C5 with a proper 5=2-coloring . 7 4 The Mycielski transformation . 14 5 The ¯rst four Zykov graphs . 22 ii Contents 1 Overview 1 2 Preliminaries 2 2.1 Graphs and graph colorings . 2 2.1.1 Basic de¯nitions . 2 2.1.2 Independent sets and cliques . 2 2.1.3 Examples . 3 2.1.4 Graph colorings and graph homomorphisms . 3 2.2 Fractional colorings and fractional cliques . 5 2.2.1 Fractional colorings . 5 2.2.2 Fractional colorings in terms of independent sets . 7 2.2.3 Fractional chromatic number as a linear program . 9 2.2.4 Fractional cliques . 10 2.2.5 Equality of ÂF and !F ..................... 11 3 Fractional Colorings and Mycielski's Graphs 13 3.1 The Mycielski transformation . 13 3.2 Main result . 14 3.2.1 Proof of part (a) . 14 3.2.2 Proof of part (b) . 15 3.2.3 Proof of part (c) . 16 3.3 Discussion of main result . 20 4 The Zykov graphs 21 Bibliography 25 1 Chapter 1 Overview In this paper, we discuss a result about graph colorings from 1995. The paper we will be investigating is \The Fractional Chromatic Number of Mycielski's Graphs," by Michael Larsen, James Propp and Daniel Ullman [3]. We will begin with some preliminary de¯nitions, examples, and results about graph colorings. Then we will de¯ne fractional colorings and the fractional chro- matic number, which are the focus of Larsen, Propp and Ullman's paper. We will de¯ne fractional colorings in two di®erent ways: ¯rst in a fairly intuitive, combi- natorial manner that is characterized in terms of graph homomorphisms, and then in terms of independent sets, which as we shall see, lends itself to calculation by means of linear programming. In this second context, we shall also de¯ne fractional cliques, and see how they relate to fractional colorings. This connection between fractional colorings and fractional cliques is the key to Larsen, Propp and Ullman's proof. 2 Chapter 2 Preliminaries 2.1 Graphs and graph colorings The material in this section is based on the exposition found in chapters 1 and 7 in the excellent book by Godsil and Royle [2]. 2.1.1 Basic de¯nitions A graph is de¯ned as a set of vertices and a set of edges joining pairs of vertices. The precise de¯nition of a graph varies from author to author; in this paper, we will consider only ¯nite, simple graphs, and shall tailor our de¯nition accordingly. A graph G is an ordered pair (V (G);E(G)), consisting of a vertex set, V (G), and an edge set, E(G). The vertex set can be any ¯nite set, as we are considering only ¯nite graphs. Since we are only considering simple graphs, and excluding loops and multiple edges, we can de¯ne E(G) as a subset of the set of all unordered pairs of distinct elements of V (G). 2.1.2 Independent sets and cliques If u and v are elements of V (G), and fu; vg 2 E(G), then we say that u and v are adjacent, denoted u » v. Adjacency is a symmetric relation, and in the case of simple graphs, anti-reflexive. A set of pairwise adjacent vertices in a graph is called a clique and a set of pairwise non-adjacent vertices is called an independent set. 3 For any graph G, we de¯ne two parameters: ®(G), the independence num- ber, and !(G), the clique number. The independence number is the size of the largest independent set in V (G), and the clique number is the size of the largest clique. 2.1.3 Examples As examples, we de¯ne two families of graphs, the cycles and the complete graphs. The cycle on n vertices (n > 1), denoted Cn, is a graph with V (Cn) = f1; : : : ; ng and x » y in Cn if and only if x ¡ y ´ §1 (mod n). We often depict Cn as a regular n-gon. The independence and clique numbers are easy to calculate: n we have ®(Cn) = b 2 c and !(Cn) = 2 (except for C3, which has a clique number of 3). The complete graph on n vertices, Kn, is a graph with V (Kn) = f1; : : : ; ng and x » y in V (Kn) for all x 6= y. It is immediate that ®(Kn) = 1 and !(Kn) = n. The graphs C5 and K5 are shown in Figure 1. Figure 1: Cycle and complete graph on ¯ve vertices 2.1.4 Graph colorings and graph homomorphisms A proper n-coloring (or simply a proper coloring) of a graph G can be thought of as a way of assigning, from a set of n \colors", one color to each vertex, in such 4 a way that no adjacent vertices have the same color. A more formal de¯nition of a proper coloring relies on the idea of graph homomorphisms. If G and H are graphs, a graph homomorphism from G to H is a mapping Á : V (G) ! V (H) such that u » v in G implies Á(u) » Á(v) in H. A bijective graph homomorphism whose inverse is also a graph homomorphism is called a graph isomorphism. Now we may de¯ne a proper n-coloring of a graph G as a graph homomorphism from G to Kn. This is equivalent to our previous, informal de¯nition, which can be seen as follows. Given a \color" for each vertex in G, with adjacent vertices always having di®erent colors, we may de¯ne a homomorphism that sends all the vertices of the same color to the same vertex in Kn. Since adjacent vertices have di®erent colors assigned to them, they will be mapped to di®erent vertices in Kn, which are adjacent. Conversely, any homomorphism from G to Kn assigns to each vertex of G an element of f1; 2; : : : ; ng, which may be viewed as colors. Since no vertex in Kn is adjacent to itself, no adjacent vertices in G will be assigned the same color. In a proper coloring, if we consider the inverse image of a single vertex in Kn, i.e., the set of all vertices in G with a certain color, it will always be an independent set. This independent set is called a color class associated with the proper coloring. Thus, a proper n-coloring of a graph G can be thought of as a covering of the vertex set of G with independent sets. We de¯ne a graph parameter Â(G), the chromatic number of G, as the small- est positive integer n such that there exists a proper n-coloring of G. Equivalently, the chromatic number is the smallest number of independent sets required to cover V (G). Any ¯nite graph with k vertices can certainly be colored with k colors, so we see that Â(G) is well-de¯ned for a ¯nite graph G, and bounded from above by jV (G)j. It is also clear that, if we have a proper n-coloring of G, then Â(G) · n. We can establish some inequalities relating the chromatic number to the other 5 parameters we have de¯ned. First, !(G) · Â(G), since all the vertices in a clique must be di®erent colors. Also, since each color class is an independent set, we jV (G)j have ®(G) · Â(G), where equality is attained if and only if each color class in an optimal coloring is the size of the largest independent set. We can calculate the chromatic number for our examples. For the complete graphs, we have Â(Kn) = n, and for the cycles we have Â(Cn) = 2 for n even and 3 for n odd. In Figure 2, we see C5 and K5 colored with three and ¯ve colors, respectively. Figure 2: The graphs C5 and K5 with proper 3- and 5-colorings, respectively 2.2 Fractional colorings and fractional cliques 2.2.1 Fractional colorings We now generalize the idea of a proper coloring to that of a fractional coloring (or a set coloring), which allows us to de¯ne a graph's fractional chromatic number, denoted ÂF (G), which can assume non-integer values. Given a graph, integers 0 < b · a, and a set of a colors, a proper a=b-coloring is a function that assigns to each vertex a set of b distinct colors, in such a way that adjacent vertices are assigned disjoint sets. Thus, a proper n-coloring is equivalent to a proper n=1-coloring. 6 The de¯nition of a fractional coloring can also be formalized by using graph homomorphisms. To this end, we de¯ne another family of graphs, the Kneser graphs. For each ordered pair of positive integers (a; b) with a ¸ b, we de¯ne a graph Ka:b. As the vertex set of Ka:b, we take the set of all b-element subsets of the set f1; : : : ; ag. Two such subsets are adjacent in Ka:b if and only if they are disjoint. Note that Ka:b is an empty graph (i.e., its edge set is empty) unless a ¸ 2b.
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