Comments on: The Chromatic Number of Liechtenstein http://bit-player.org/2008/the-chromatic-number-of-liechtenstein An amateur's outlook on computation and mathematics. Tue, 16 Mar 2010 08:41:37 +0000 http://wordpress.org/?v=2.6.3 By: Cahit http://bit-player.org/2008/the-chromatic-number-of-liechtenstein#comment-1835 Cahit Mon, 03 Nov 2008 07:30:42 +0000 http://bit-player.org/?p=191#comment-1835 The Hadwiger's conjecture which asserts that every loopless graph not contractible to the complete graph on k + 1 vertices is k-colorable. When k = 3 this is easy, and when k= 4, Wagner's theorem of 1937 shows the conjecture to be equivalent to the four-color theorem (the 4CT)[1]. The case k = 5 it is also equivalent to the 4CT. Without assuming the 4CT, Robertson, Seymour and Thomas have shown that every minimal counterexample to Hadwiger's conjecture, when k = 5 is apex ; that is, it consists of a planar graph with one additional vertex violating the planarity. Consequently, the 4CT implies Hadwiger's conjecture when k = 5, because it implies that apex graphs are 5-colorable [2]. For other values of k the Hadwiger conjecture is open. Therefore the general graph coloring problem related with the Liechtenstein map is rely on the solution of the Hadwiger's conjecture. But here we have the option to consider equivalently, coloring of the dual planar graph. [1] K. Wagner, Uber eine Eigenschaft der ebenen Komplexe", Math. Ann. 114 (1937),570-590. [2] N. Robertson, P. Seymour, R. Thomas, Hadwiger's conjecture for K6-free graphs, Combinatorica 14 (1993), 279-361. The Hadwiger’s conjecture which asserts that every loopless graph not contractible to the complete graph on k + 1 vertices is k-colorable. When k = 3 this is easy, and when k= 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the four-color theorem (the 4CT)[1]. The case k = 5 it is also equivalent to the 4CT. Without assuming the 4CT, Robertson, Seymour and Thomas have shown that every minimal counterexample to Hadwiger’s conjecture, when k = 5 is apex ; that is, it consists of a planar graph with one additional vertex violating the planarity. Consequently, the 4CT implies Hadwiger’s conjecture when k = 5, because it implies that apex graphs are 5-colorable [2]. For other values of k the Hadwiger conjecture is open. Therefore the general graph coloring problem related with the Liechtenstein map is rely on the solution of the Hadwiger’s conjecture. But here we have the option to consider equivalently, coloring of the dual planar graph.

[1] K. Wagner, Uber eine Eigenschaft der ebenen Komplexe”, Math. Ann. 114 (1937),570-590.

[2] N. Robertson, P. Seymour, R. Thomas, Hadwiger’s conjecture for K6-free graphs, Combinatorica 14 (1993), 279-361.

]]> By: Mikael Vejdemo Johansson http://bit-player.org/2008/the-chromatic-number-of-liechtenstein#comment-1834 Mikael Vejdemo Johansson Mon, 03 Nov 2008 05:59:13 +0000 http://bit-player.org/?p=191#comment-1834 What is the status on Hadwiger's conjecture? What is the status on Hadwiger’s conjecture?

]]> By: Cahit http://bit-player.org/2008/the-chromatic-number-of-liechtenstein#comment-1833 Cahit Sat, 01 Nov 2008 06:33:12 +0000 http://bit-player.org/?p=191#comment-1833 The map coloring problem given by the chromatic number of Liechtenstein is quite interesting from the point of graph coloring. You may deal with the constraint planar graph coloring problem if you only consider dual-graph of the map. Here constraint comes from the coloring of the nodes of the disjoint regions corresponding to the same commune with the same color. Equivalently you can study coloring of the map by coloring the nodes of the non-planar graph obtained by shrinking all regions of a commune to a single node (see Michi's Blog). Then the coloring problem can be handled by the Hadwiger's conjecture which says "every K_k-minor free graph is colorable with k-1 colors. I also note that one can design map of this kind for which the size of the maximum clique corresponding to the non-planar graph is equals to the number of communes. The map coloring problem given by the chromatic number of Liechtenstein is quite interesting from the point of graph coloring. You may deal with the constraint planar graph coloring problem if you only consider dual-graph of the map. Here constraint comes from the coloring of the nodes of the disjoint regions corresponding to the same commune with the same color. Equivalently you can study coloring of the map by coloring the nodes of the non-planar graph obtained by shrinking all regions of a commune to a single node (see Michi’s Blog). Then the coloring problem can be handled by the Hadwiger’s conjecture which says “every K_k-minor free graph is colorable with k-1 colors. I also note that one can design map of this kind for which the size of the maximum clique corresponding to the non-planar graph is equals to the number of communes.

]]> By: brian http://bit-player.org/2008/the-chromatic-number-of-liechtenstein#comment-1832 brian Fri, 31 Oct 2008 19:36:07 +0000 http://bit-player.org/?p=191#comment-1832 There's still more at <a href="http://blog.mikael.johanssons.org/" rel="nofollow">Michi's Blog</a>. In addition, Ibrahim Cahit has produced <a href="http://www.flickr.com/photos/49058045@N00/2985004680/" rel="nofollow">an explicit five-coloring</a> of the map itself, complete with Blackletter labels. There’s still more at Michi’s Blog. In addition, Ibrahim Cahit has produced an explicit five-coloring of the map itself, complete with Blackletter labels.

]]> By: Mikael Vejdemo Johansson http://bit-player.org/2008/the-chromatic-number-of-liechtenstein#comment-1830 Mikael Vejdemo Johansson Tue, 28 Oct 2008 18:42:29 +0000 http://bit-player.org/?p=191#comment-1830 There is a five-coloring. See my blog. I'm trying not to spoil too much in here. There is a five-coloring. See my blog.

I’m trying not to spoil too much in here.

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