Hadwiger’s Conjecture and Seagull Packing

T he four-color theorem [1, 2] is one of the most well-known results in graph theory. Originating from the question of coloring a world map, posed in the middle of the nineteenth century, it has since fascinated hundreds of researchers and motivated a lot of beautiful mathematics. It states that every planar graph (that is, a graph that can be drawn in the plane without crossings) can be properly colored with four colors. Kuratowski’s theorem [15] connects the planarity of a graph with the absence of certain topological structures in it. Thus, the four-color theorem tells us that if a graph lacks a certain topological obstruction, then it is four-colorable. In 1941 Hadwiger made a conjecture that generalized this idea, from four-colorability to general t-colorability. Since then, Hadwiger’s conjecture has received a great deal of attention, but only a few special cases have been solved. In this article we survey some of the known results on the conjecture and discuss recent progress. To make this more precise, let us say that a graph G contains a Kt -minor, or a clique minor of size t , if there exist t pairwise vertex-disjoint connected subgraphs F1, . . . , Ft of G, such that for every distinct i, j ∈ {1, . . . t}, there is an edge with one end in V(Fi) and the other in V(Fj). (For a graph F , we denote by V(F) the vertex set of F , and by E(F) the edge set of F .) For coloring problems, it is necessary to assume that graphs are loopless, that is, every edge has two distinct ends, and we adopt this convention here. We can now state Hadwiger’s conjecture [10].