Brooks' theorem
{{Short description|On graph coloring and neighborhood size}}
File:Graph exact coloring.svgs need one more color than their maximum degree. They and the odd cycles are the only exceptions to Brooks' theorem.]]
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.{{sfnp|Brooks|1941}} A coloring with the number of colors described by Brooks' theorem is sometimes called a Brooks coloring{{sfnp|Hajnal|Szemerédi|1990}} or a Δ-coloring.{{sfnp|Panconesi|Srinivasan|1995}}
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.{{sfnp|Brooks|1941}}
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.{{sfnp|Lovász|1975}}
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.{{sfnp|Vizing|1976}}
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.{{sfnp|Reed|1999}} For triangle-free graphs, or more generally graphs in which the neighborhood of every vertex is sufficiently sparse, O(Δ/log Δ) colors suffice.{{harvtxt|Alon|Krivelevich|Sudakov|1999}}.
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.{{harvtxt|Skulrattanakulchai|2006}}. Efficient algorithms are also known for finding Brooks colorings in parallel and distributed models of computation.{{harvtxt|Karloff|1989}}; {{harvtxt|Hajnal|Szemerédi|1990}}; {{harvtxt|Panconesi|Srinivasan|1995}}; {{harvtxt|Grable|Panconesi|2000}}.
Notes
{{reflist|25em}}
References
{{refbegin|25em}}
- {{citation
| last1 = Alon | first1 = Noga | author1-link = Noga Alon
| last2 = Krivelevich | first2 = Michael | author2-link = Michael Krivelevich
| last3 = Sudakov | first3 = Benny | author3-link = Benny Sudakov
| doi = 10.1006/jctb.1999.1910 |doi-access=free
| issue = 1
| journal = Journal of Combinatorial Theory | series = Series B
| pages = 73–82
| title = Coloring graphs with sparse neighborhoods
| volume = 77
| year = 1999}}
- {{citation
| last = Brooks | first = R. L. | authorlink = R. Leonard Brooks
| journal = Mathematical Proceedings of the Cambridge Philosophical Society
| pages = 194–197
| title = On colouring the nodes of a network
| volume = 37
| doi = 10.1017/S030500410002168X
| year = 1941| issue = 2 | bibcode = 1941PCPS...37..194B | s2cid = 209835194 }}.
- {{citation
| last1 = Grable | first1 = David A.
| last2 = Panconesi | first2 = Alessandro
| title = Fast distributed algorithms for Brooks–Vizing colourings
| pages = 85–120
| volume = 37
| journal = Journal of Algorithms
| doi = 10.1006/jagm.2000.1097
| year = 2000| s2cid = 14211416
}}.
- {{citation
| last1 = Hajnal | first1 = Péter
| last2 = Szemerédi | first2 = Endre | author2-link = Endre Szemerédi
| doi = 10.1137/0403008
| issue = 1
| journal = SIAM Journal on Discrete Mathematics
| pages = 74–80
| title = Brooks coloring in parallel
| volume = 3
| year = 1990}}.
- {{citation
| last = Karloff | first = H. J.
| doi = 10.1016/0304-3975(89)90121-7
| issue = 1
| journal = Theoretical Computer Science
| pages = 89–103
| title = An NC algorithm for Brooks' theorem
| volume = 68
| year = 1989| doi-access =
}}.
- {{citation
| last = Lovász | first = L. | author-link = László Lovász
| doi = 10.1016/0095-8956(75)90089-1
| journal = Journal of Combinatorial Theory | series = Series B
| pages = 269–271
| title = Three short proofs in graph theory
| volume = 19
| year = 1975| issue = 3 | doi-access = free
}}.
- {{citation
| last1 = Panconesi | first1 = Alessandro
| last2 = Srinivasan | first2 = Aravind
| doi = 10.1007/BF01200759
| issue = 2
| journal = Combinatorica
| pages = 255–280
| title = The local nature of Δ-coloring and its algorithmic applications
| volume = 15
| year = 1995| s2cid = 28307157
}}.
- {{citation
| doi = 10.1006/jctb.1998.1891
| last = Reed | first = Bruce | authorlink = Bruce Reed (mathematician)
| issue = 2
| journal = Journal of Combinatorial Theory | series = Series B
| pages = 136–149
| title = A strengthening of Brooks' theorem
| volume = 76
| year = 1999| doi-access = free
}}.
- {{citation
| last = Skulrattanakulchai | first = San
| doi = 10.1016/j.ipl.2005.12.007
| issue = 3
| journal = Information Processing Letters
| pages = 101–106
| title = Δ-List vertex coloring in linear time
| volume = 98
| year = 2006}}.
- {{citation|last=Vizing|first= V. G.|year=1976|title=Vertex colorings with given colors|language=Russian|journal=Diskret. Analiz.|volume=29|pages=3–10}}.
{{refend}}
External links
- {{mathworld|title=Brooks' Theorem|urlname=BrooksTheorem|mode=cs2}}