In graph theory, Brooks' theorem states a relationship between the maximum degree of a graph and its chromatic number. According to the theorem, in a connected graph in which every vertex has at most ÃÂ neighbors, the vertices can be colored with only ÃÂ colors, except for two cases, complete graphs and cycle graphs of odd length, which require ÃÂ + 1 colors.
The theorem is named after R. Leonard Brooks, who published a proof of it in 1941. A coloring with the number of colors described by Brooks' theorem is sometimes called a Brooks coloring or a ÃÂ-coloring.
For any connected undirected graph G with maximum degree ÃÂ, the chromatic number of G is at most ÃÂ, unless G is a complete graph or an odd cycle, in which case the chromatic number is ÃÂ + 1.
László Lovász gives a simplified proof of Brooks' theorem. If the graph is not biconnected, its biconnected components may be colored separately and then the colorings combined. If the graph has a vertex v with degree less than ÃÂ, then a greedy coloring algorithm that colors vertices farther from v before closer ones uses at most àcolors. This is because at the time that each vertex other than v is colored, at least one of its neighbors (the one on a shortest path to v) is uncolored, so it has fewer than àcolored neighbors and has a free color. When the algorithm reaches v, its small number of neighbors allows it to be colored. Therefore, the most difficult case of the proof concerns biconnected ÃÂ-regular graphs with àâÂÂ¥ 3. In this case, Lovász shows that one can find a spanning tree such that two nonadjacent neighbors u and w of the root v are leaves in the tree. A greedy coloring starting from u and w and processing the remaining vertices of the spanning tree in bottom-up order, ending at v, uses at most àcolors. For, when every vertex other than v is colored, it has an uncolored parent, so its already-colored neighbors cannot use up all the free colors, while at v the two neighbors u and w have equal colors so again a free color remains for v itself.
A more general version of the theorem applies to list coloring: given any connected undirected graph with maximum degree ÃÂ that is neither a clique nor an odd cycle, and a list of ÃÂ colors for each vertex, it is possible to choose a color for each vertex from its list so that no two adjacent vertices have the same color. In other words, the list chromatic number of a connected undirected graph G never exceeds ÃÂ, unless G is a clique or an odd cycle.
For certain graphs, even fewer than ÃÂ colors may be needed. ÃÂ − 1 colors suffice if and only if the given graph has no ÃÂ-clique, provided ÃÂ is large enough. For triangle-free graphs, or more generally graphs in which the neighborhood of every vertex is sufficiently sparse, O(ÃÂ/log ÃÂ) colors suffice.
The degree of a graph also appears in upper bounds for other types of coloring; for edge coloring, the result that the chromatic index is at most à+ 1 is Vizing's theorem. An extension of Brooks' theorem to total coloring, stating that the total chromatic number is at most à+ 2, has been conjectured by Mehdi Behzad and Vizing. The HajnalâÂÂSzemerédi theorem on equitable coloring states that any graph has a (à+ 1)-coloring in which the sizes of any two color classes differ by at most one.
A ÃÂ-coloring, or even a ÃÂ-list-coloring, of a degree-ÃÂ graph may be found in linear time. Efficient algorithms are also known for finding Brooks colorings in parallel and distributed models of computation.