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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 Δ neighbors, the vertices can be colored with only Δ colors, except for two cases, complete graphs and cycle graphs of odd length, which require Δ + 1 colors.

The theorem is named after R. Leonard Brooks, who published a proof of it in 1941.^[1] A coloring with the number of colors described by Brooks' theorem is sometimes called a Brooks coloring^[2] or a Δ-coloring.^[3]

Formal statement

For any connected undirected graph G with maximum degree Δ, the chromatic number of G is at most Δ, unless G is a complete graph or an odd cycle, in which case the chromatic number is Δ + 1.^[1]

Proof

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 Δ, then a greedy coloring algorithm that colors vertices farther from v before closer ones uses at most Δ colors. 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 Δ colored 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 Δ-regular graphs with Δ ≥ 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 Δ colors. 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.^[4]

Extensions

A more general version of the theorem applies to list coloring: given any connected undirected graph with maximum degree Δ that is neither a clique nor an odd cycle, and a list of Δ 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 Δ, unless G is a clique or an odd cycle.^[5]

For certain graphs, even fewer than Δ colors may be needed. Δ − 1 colors suffice if and only if the given graph has no Δ-clique, provided Δ is large enough.^[6] For triangle-free graphs, or more generally graphs in which the neighborhood of every vertex is sufficiently sparse, O(Δ/log Δ) colors suffice.^[7]

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 Δ + 1 is Vizing's theorem. An extension of Brooks' theorem to total coloring, stating that the total chromatic number is at most Δ + 2, has been conjectured by Mehdi Behzad and Vizing. The Hajnal–Szemerédi theorem on equitable coloring states that any graph has a (Δ + 1)-coloring in which the sizes of any two color classes differ by at most one.

Algorithms

A Δ-coloring, or even a Δ-list-coloring, of a degree-Δ graph may be found in linear time.^[8] Efficient algorithms are also known for finding Brooks colorings in parallel and distributed models of computation.^[9]

Notes

References

Alon, Noga; Krivelevich, Michael; Sudakov, Benny (1999), "Coloring graphs with sparse neighborhoods", Journal of Combinatorial Theory, Series B, 77 (1): 73–82, doi:10.1006/jctb.1999.1910
Brooks, R. L. (1941), "On colouring the nodes of a network", Mathematical Proceedings of the Cambridge Philosophical Society, 37 (2): 194–197, Bibcode:1941PCPS...37..194B, doi:10.1017/S030500410002168X, S2CID 209835194.
Grable, David A.; Panconesi, Alessandro (2000), "Fast distributed algorithms for Brooks–Vizing colourings", Journal of Algorithms, 37: 85–120, doi:10.1006/jagm.2000.1097, S2CID 14211416.
Hajnal, Péter; Szemerédi, Endre (1990), "Brooks coloring in parallel", SIAM Journal on Discrete Mathematics, 3 (1): 74–80, doi:10.1137/0403008.
Karloff, H. J. (1989), "An NC algorithm for Brooks' theorem", Theoretical Computer Science, 68 (1): 89–103, doi:10.1016/0304-3975(89)90121-7.
Lovász, L. (1975), "Three short proofs in graph theory", Journal of Combinatorial Theory, Series B, 19 (3): 269–271, doi:10.1016/0095-8956(75)90089-1.
Panconesi, Alessandro; Srinivasan, Aravind (1995), "The local nature of Δ-coloring and its algorithmic applications", Combinatorica, 15 (2): 255–280, doi:10.1007/BF01200759, S2CID 28307157.
Reed, Bruce (1999), "A strengthening of Brooks' theorem", Journal of Combinatorial Theory, Series B, 76 (2): 136–149, doi:10.1006/jctb.1998.1891.
Skulrattanakulchai, San (2006), "Δ-List vertex coloring in linear time", Information Processing Letters, 98 (3): 101–106, doi:10.1016/j.ipl.2005.12.007.
Vizing, V. G. (1976), "Vertex colorings with given colors", Diskret. Analiz. (in Russian), 29: 3–10.