The chromatic number of a graph 
is the smallest number of colors needed to color the vertices
of 
so that no two adjacent vertices share the same color (Skiena 1990, p. 210),
i.e., the smallest value of 
possible to obtain a k-coloring.
Minimal colorings and chromatic numbers for a sample of graphs are illustrated above.
The chromatic number of a graph 
is most commonly denoted 
(e.g., Skiena 1990, West 2000, Godsil and Royle 2001,
Pemmaraju and Skiena 2003), but occasionally also 
.
Empty graphs have chromatic number 1, while non-empty bipartite graphs have chromatic number 2.
The chromatic number of a graph 
is also the smallest positive integer 
such that the chromatic
polynomial 
.
Calculating the chromatic number of a graph is an NP-complete
problem (Skiena 1990, pp. 211-212). Or, in the words of Harary (1994, p. 127),
"no convenient method is known for determining the chromatic number of an arbitrary
graph." However, Mehrotra and Trick (1996) devised a column generation algorithm
for computing chromatic numbers and vertex colorings which solves most small to moderate-sized
graph quickly.
Computation of the chromatic number of a graph is implemented in the Wolfram Language as VertexChromaticNumber[g]. Precomputed chromatic numbers for many named graphs can be obtained using GraphData[graph, "ChromaticNumber"].
The chromatic number of a graph must be greater than or equal to its clique number. A graph is called a perfect graph if,
for each of its induced subgraphs 
, the chromatic number of 
equals the largest number of pairwise adjacent vertices
in 
.
A graph for which the clique number is equal to
the chromatic number (with no further restrictions on induced subgraphs) is said
to be weakly perfect.
By definition, the edge chromatic number of a graph 
equals the chromatic number of the line graph 
.
Brooks' theorem states that the chromatic number of a graph is at most the maximum vertex degree 
, unless the graph is complete
or an odd cycle, in which case 
colors are required.
A graph with chromatic number 
is said to be bicolorable,
and a graph with chromatic number 
is said to be three-colorable.
In general, a graph with chromatic number 
is said to be an k-chromatic
graph, and a graph with chromatic number 
is said to be k-colorable.
The following table gives the chromatic numbers for some named classes of graphs.
, 
, 
, 
For any two positive integers 
and 
, there exists a graph of girth at least 
and chromatic number at least 
(Erdős 1961; Lovász 1968; Skiena 1990, p. 215).
The chromatic number of a surface of genus 
is given by the Heawood
conjecture,
 |
where 
is the floor function. 
is sometimes also denoted 
(which is unfortunate, since 
commonly refers to the Euler
characteristic). For 
, 1, ..., the first few values of 
are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16,
... (OEIS A000934).
Erdős (1959) proved that there are graphs with arbitrarily large girth and chromatic number (Bollobás and West 2000).
