Abstract
List coloring is a generalization of graph coloring introduced by Erdős, Rubin and Taylor in 1980, which has become extensively studied in graph theory. A graph G is said to be k-choosable, or k-list-colorable, if, for every way of assigning a list (set) of k colors to each vertex of G, it is possible to choose a color from each list in such a way that no two neighboring vertices receive the same color. Note that if the lists are all the same, then this is asking for G to have chromatic number at most k. One might think that the case where all the lists are the same would be the hardest: surely making the lists different should make it easier to ensure that neighboring vertices have different colors. Rather surprisingly, however, this is not the case. A counterexample is provided by the complete bipartite graph K2,4. If the two vertices in the first vertex class are assigned the lists {a,b} and {c,d}, while the vertices in the other vertex class are assigned the lists {a,c}, {a,d}, {b,c} and {b,d}, then it is easy to check that it is not possible to obtain a proper coloring from these lists, so G is not 2-choosable, and yet the chromatic number of G is 2. A famous theorem of Galvin, which solved the so-called Dinitz conjecture, states that the line graph of the complete bipartite graph Kn,n is n-choosable. Equivalently, if each square of an n×n grid is assigned a list of n colors, it is possible to choose a color from each list in such a way that no color appears more than once in any row or column. One can generalize this notion by requiring a choice of not just one color from each list, but some larger number of colors. A graph G is said to be (A,B)-list-colorable if, for every assignment of lists to the vertices of G, each consisting of A colors, there is an assignment of sets of B colors to the vertices such that each vertex is assigned a set that is a subset of its list and the sets assigned to pairs of adjacent vertices are disjoint. (When B=1 this simply says that G is A-choosable.) In this short paper, the authors answer a question that has remained open for almost four decades since it was posed by Erdős, Rubin and Taylor in their seminal paper: if a graph is (A,B)-list-colorable, is it true that it is also (mA,mB)-list-colorable for every m≥1? Quite surprisingly, the answer is again negative - the authors construct a graph that is (4,1)-list-colorable but not (8,2)-list-colorable.
Highlights
Coloring the vertices of a graph with sets of colors is a fundamental notion, which in particular captures fractional colorings
A function that assigns a set to each vertex of a graph is a set coloring if the sets assigned to adjacent vertices are disjoint
For positive integers a and b ≤ a, an (a : b)-coloring of a graph G is a set coloring with range
Summary
Coloring the vertices of a graph with sets of colors (that is, each vertex is assigned a fixed-size subset of the colors such that adjacent vertices are assigned disjoint sets) is a fundamental notion, which in particular captures fractional colorings. The fractional chromatic number of a graph G can be defined to be the infimum (which is a minimum) of the ratios a/b such that, if every vertex of G is replaced by a clique of order b and every edge of G is replaced by a complete bipartite graph between the relevant cliques, the chromatic number of the obtained graph is at most a. In their seminal work on list coloring, Erdos, Rubin and Taylor [2] raised several intriguing questions about the list version of set coloring. Gutner and Tarsi [3] demonstrated that if k and m are positive integers and k is odd, every (2mk : mk)-choosable graph is 2m-choosable
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