Hajnal–Szemerédi theorem
In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that
That is, the partition of vertices among the different colors is as uniform as possible. For instance, giving each vertex a distinct color would be equitable, but would typically use many more colors than are necessary in an optimal equitable coloring. An equivalent way of defining an equitable coloring is that it is an embedding of the given graph as a subgraph of a Turán graph with the same set of vertices. There are two kinds of chromatic number associated with equitable coloring. The equitable chromatic number of a graph G is the smallest number k such that G has an equitable coloring with k colors. But G might not have equitable colorings for some larger numbers of colors; the equitable chromatic threshold of G is the smallest k such that G has equitable colorings for any number of colors greater than or equal to k.
The Hajnal–Szemerédi theorem, posed as a conjecture by Paul Erdős (1964) and proven by András Hajnal and Endre Szemerédi (1970), states that any graph with maximum degree Δ has an equitable coloring with Δ + 1 colors. Several related conjectures remain open. Polynomial time algorithms are also known for finding a coloring matching this bound, and for finding optimal colorings of special classes of graphs, but the more general problem of deciding whether an arbitrary graph has an equitable coloring with a given number of colors is NP-complete.
The star K1,5 shown in the illustration is a complete bipartite graph, and therefore may be colored with two colors. However, the resulting coloring has one vertex in one color class and five in another, and is therefore not equitable. The smallest number of colors in an equitable coloring of this graph is four, as shown in the illustration: the central vertex must be the only vertex in its color class, so the other five vertices must be split among at least three color classes in order to ensure that the other color classes all have at most two vertices. More generally, Meyer (1973) observes that any star K1,n needs colors in any equitable coloring; thus, the chromatic number of a graph may differ from its equitable coloring number by a factor as large as n/4. Because K1,5 has maximum degree five, the number of colors guaranteed for it by the Hajnal–Szemerédi theorem is six, achieved by giving each vertex a distinct color.
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