1. A graph is a set of vertices and edges between some vertices.. A bipartite graph has two clumps of vertices, with edges going from a vertex in one clump to a vertex in the other clump. Given a graph and a set of colors, a proper q-coloring is a coloring of the vertices using q colors so that any two vertices joined by an edge are given different colors. 4-coloring of a bipartite graph Graph colorings show up many different contexts, such as coloring maps, assigning time slots to presentations at a conference, and solutions to puzzles like Sudoku. We study the bipartite graph called the d-dimensional discrete hypercube, for a positive integer d. This is a graph whose vertices are given by each length d string composed with 0’s and 1’s; there are 2 d vertices. Edges join two strings which differ in exactly one coordinate. 3-dimensional hypercube 3-dimensional hypercube (0-1 strings shown) All results hold asymptotically as d gets large. Result What’s Next? The Typical Structure of Proper Colorings of the Discrete Hypercube John Engbers and David Galvin Department of Mathematics, University of Notre Dame Bibliography We want to answer the basic question: If you select a random proper coloring of the discrete hypercube, what does it look like? A simple method used to color the vertices is to split the colors in half ({R,Y},{B,G}) and use {R,Y} on one partition class and {B,G} on the other partition class. Simple 4-coloring Simple 5-coloring In fact, most colorings look similar to simple colorings: Theorem (E., Galvin 2011+): In a random (uniformly chosen) proper q-coloring of the discrete hypercube, almost all appearances of each color are on one bipartition class. Corollary: The space of all proper 5-colorings of the hypercube splits into 20 large classes, based on the dominant colors in one partition class, and one small extra class. Corollary: The discrete hypercube exhibits long-range influence. 1.Chung, F.; Frankl, P.; Graham, R. and Shearer, J., Some intersection theorems for ordered sets and graphs, Journal of Combinatorial Theory Series A, Vol 48 (1986), 23-37. 2.Engbers, J. and Galvin, D. H-coloring tori, Submitted. 3.Engbers, J. and Galvin, D. H-coloring bipartite graphs, Submitted. 4.Galvin, D. On homomorphisms from the Hamming cube to Z, Israel Journal of Mathematics, Vol 138 (2003), 189-213. 5.Kahn, J. Range of cube-indexed random walk, Israel Journal of Mathematics, Vol 124 (2001), 189-201. 6.Kahn, J. An entropy approach to the hard-core model on bipartite graphs, Combinatorics, Probability, & Computing, Vol 10 (2001) 219-237. The proof generalizes an idea of Kahn ([5]): Local Structure: Assume: red vertices are colored from set of colors A, blue vertices are colored from set of colors B. We maximize:, the number of ways to color x, y, and a pair of red/blue vertices, under this assumption. This is maximized by an A and B expected from the simple method coloring. Local to global : we use the notion of entropy. Briefly, for a random variable X, the entropy H(X) is the function:. Entropy’s relationship with counting:. Shearer’s lemma ([1]) completes local to global:. Generalizations: We’ve proved this result for the more general discrete even torus and generalized colorings (graph homomorphisms to a fixed target graph H) in [2]. These results generalize some of the results in [6]. A few of the basic structural questions for any bipartite graph, not just the discrete hypercube, are answered in [3]. New Questions: 1. How many q-colorings of the d-dimensional discrete hypercube are there? - 2-colorings: 2 - 3-colorings ([4]): ~ 6e2 2 d-1 - 4-, 5-, 6-, … colorings: ??? 2. Run bond percolation on the discrete hypercube. Is there a threshold for obtaining similar structural results? 3. Does Glauber dynamics on the space of proper q- colorings of the discrete hypercube exhibit slow mixing times? 4. Does the infinite lattice exhibit long-range influence? Introduction Idea of the Proof Poster template from www.postersession.com. Printed at the University of Notre Dame.www.postersession.com