ISSN:
1439-6912
Keywords:
05 C 15
;
05 C 75
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01202354
