Hadwiger's Conjecture for the Complements of Kneser Graphs

A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. An H-minor is a minor isomorphic to H. The Hadwiger number of G, denoted by h(G), is the maximum integer t such that G contains a Kt-minor, where Kt is the complete graph with t vertices. Hadwiger [8] conjectured that every graph that is not (t−1)-colourable contains a Kt-minor; that is, h(G) ≥ χ(G) for every graph G, where χ(G) is the chromatic number of G. Hadwiger’s conjecture is widely believed to be one of the most difficult and beautiful problems in graph theory. It has been proved [11] for graphs with χ(G) ≤ 6, and is open for graphs with χ(G) ≥ 7. It has also been proved for certain special classes of graphs, including powers of cycles and their complements [9], proper circular arc graphs [2], line graphs [10], quasi-line graphs [6] and 3-arc graphs [7]. See [13] for a survey. A strengthening of Hadwiger’s conjecture due to Hajós asserts that every graph G with χ(G) ≥ t contains a subdivision of Kt. Catlin [4] proved that Hajós’ conjecture fails for every t ≥ 7. Obviously, if Hadwiger’s conjecture is false, then counterexamples must be found among counterexamples to Hajós’ conjecture. In [12] Thomassen presented several new classes of counterexamples to Hajós’ conjecture, including the complements of the Kneser graphs K(3k− 1, k) for sufficiently large k. (The Kneser graph K(n, k) is the graph with vertices the k-subsets of an n-set such that two vertices are adjacent if and only if the corresponding k-subsets are disjoint.) He wrote [12] that ‘it does not seem obvious’ that these classes all satisfy Hadwiger’s conjecture. Motivated by this comment, we prove in this paper that indeed the complement of every Kneser graph satisfies Hadwiger’s conjecture. We notice that in the special case when k divides n this was established in [9].