Exact Graph Coloring Using Inclusion-Exclusion

Years and Authors of Summarized Original Work

• 2006; Björklund, Husfeldt

Problem Definition

A k-coloring of a graph G = (V, E) assigns one of k colors to each vertex such that neighboring vertices have different colors. This is sometimes called vertex coloring.

The smallest integer k for which the graph G admits a k-coloring is denoted χ(G) and called the chromatic number. The number of k-colorings of G is denoted P(G; k) and called the chromatic polynomial.

Key Results

The central observation is that χ(G) and P(G; k) can be expressed by an inclusion-exclusion formula whose terms are determined by the number of independent sets of induced subgraphs of G. For $X \subseteq V$, let s(X) denote the number of nonempty independent vertex subsets disjoint from X, and let s _r (X) denote the number of ways to choose r nonempty independent vertex subsets S₁, …, S _r (possibly overlapping and with repetitions), all disjoint from X, such that \(\vert S_{1}\vert + \cdots +\vert S_{r}\vert =\vert V...