List coloring

In graph theory, a branch of mathematics, list coloring is a type of graph coloring where each vertex can be restricted to a list of allowed colors. It was first studied in the 1970s in independent papers by Vizing and by Erdős, Rubin, and Taylor. Given a graph G and given a set L(v) of colors for each vertex v (called a list), a list coloring is a choice function that maps every vertex v to a color in the list L(v). As with graph coloring, a list coloring is generally assumed to be proper, meaning no two adjacent vertices receive the same color. A graph is k-choosable (or k-list-colorable) if it has a proper list coloring no matter how one assigns a list of k colors to each vertex. The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable. More generally, for a function f assigning a positive integer f(v) to each vertex v, a graph G is f-choosable (or f-list-colorable) if it has a list coloring no matter how one assigns a list of f(v) colors to each vertex v. In particular, if for all vertices v, f-choosability corresponds to k-choosability. Consider the complete bipartite graph G = K2,4, having six vertices A, B, W, X, Y, Z such that A and B are each connected to all of W, X, Y, and Z, and no other vertices are connected. As a bipartite graph, G has usual chromatic number 2: one may color A and B in one color and W, X, Y, Z in another and no two adjacent vertices will have the same color. On the other hand, G has list-chromatic number larger than 2, as the following construction shows: assign to A and B the lists {red, blue} and {green, black}. Assign to the other four vertices the lists {red, green}, {red, black}, {blue, green}, and {blue, black}. No matter which choice one makes of a color from the list of A and a color from the list of B, there will be some other vertex such that both of its choices are already used to color its neighbors. Thus, G is not 2-choosable.