Hadwiger's theorem

Abstract: In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph ont+1 vertices ist-colourable. Whent≤3 this is easy, and whent=4, Wagner's theorem of 1937 shows the conjecture to be equivalent to the four-colour conjecture (the 4CC). However, whent≥5 it has remained open. Here we show that whent=5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger's conjecture whent=5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger's conjecture whent=5, because it implies that apex graphs are 5-colourable.

Abstract: One of the most beautiful and important results in geometric convexity is Hadwiger's characterization theorem for the quermassintegrals. Hadwiger's theorem classifies all continuous rigid motion invariant valuations on convex bodies as consisting of the linear span of the quermassintegrals (or, equivalently, of the intrinsic volumes) [4]. Hadwiger's characterization leads to effortless proofs of numerous results in integral geometry, including various kinematic formulas [7, 9] and the mean projection formulas for convex bodies [10]. Hadwiger's result also provides a connection between rigid motion invariant set functions and symmetric polynomials [1, 7].

Abstract: All upper semicontinuous and SL(n) invariant valuations on convex bodies containing the origin in their interiors are completely classified. Each such valuation is shown to be a linear combination of the Euler characteristic, the volume, the volume of the polar body, and the recently discovered Orlicz surface areas.

Abstract: This chapter reviews Hadwiger's number and the stability number. Hadwiger's conjecture is proved in the chapter, along with a review of the theorem related to it.