Mixing Numbers and Unfriendly Colorings of Graphs

Abstract

We only consider vertex colorings, so a "colored graph" always means a vertex-colored graph. An n-coloring of a graph is a partition of the vertices into n disjoint subsets. We start with 2-colorings; call the colors red and blue. Two vertices are neighbors if they are connected by an edge. We say that two vertices of the same color are friends and two vertices of opposite colors are strangers. If more than half the neighbors of a colored vertex v are friends of v, we say that v lives in a friendly neighborhood; otherwise, v is said to live in an unfriendly neighborhood. If all the vertices of the graph have the same color, every vertex lives in a friendly neighborhood. Is there a 2-coloring such that every vertex lives in an unfriendly neighborhood? The surprising answer to this question is yes, as we shall show. A 2-coloring of a graph is unfriendly if each vertex lives in an unfriendly neighborhood, that is, at least half its neighbors are colored differently from itself. It is a theorem that every finite graph has an unfriendly coloring. (The situation is much more complicated for infinite graphs [1, 2]). The proof is clever, but not very long and we give it next. Define the mixing number of a colored graph to be the number of its edges whose vertices have different colors. Proceed by successively "flipping," that is, changing the color of those vertices that live in friendly neighborhoods. When a vertex is flipped, it may change the neighborhood status of other vertices; however, each flip increases the mixing number of the graph. Since the mixing number is bounded by the number of edges in the graph, this flipping process must eventually end with no more flippable vertices, that is, no more vertices living in friendly neighborhoods.