The Four Colour Theorem proves that no more than four colours are required to colour the regions of any map in such a way that no two adjacent regions have the same colour.
While not exactly related to the math content, I was interested in the Siciliy map, partly because I've always been fascinated by the four color theorem but also becasue I was recently there. Last summer, I visited Siciliy with friends, and made a point of climbing to the summit while I was there. As we sat at an outside bar in the evening with a view of the mountain, it erupted. It has been much more active this year, but I caught it just as it was starting to get more active. I'm sorry to say I hadn't appreciated the map implications at the time, but I do now. Thanks.
Not that it is not true that the number of colors is bounded. This is a big open question. The main conjecture is that if ≤q countries are allowed to share a point, then the number of colors needed is at most 3/2 q. This is true when q=3 by the Four Color Theorem. And Borodin proved it when q=4. But it is not proved when q=5. Oh: It has been proved also when q=6.
I omitted the word “planar” when quoting your theorem. Is that the mistake?
In your paper, you quote a result by Ore and Plummer that says that the chromatic number of the facial closure is bounded above by 2q. Doesn’t that prove that the number of colours is “bounded as a function of q”? I thought that the big open question was that the optimal bound is 3q/2.
You are right about the 2q bound. I thought you were referring to 3/2 q, which you were not. So there is no error, except that, yes, planar must be mentioned. Note that the dual of a map is generally called a graph. But it is not wrong to call it a map. We have just submitted our work for publication, so we will see what editor/referees say. It was exciting to find the Sicilian anomaly.
There are 10 towns in N Ireland that meet at a point. See the wikipedia entry on "Quadripoints" where one finds: In Northern Ireland, ten townlands meet at the summit of Knocklayd.[62][63][64] The townlands are, clockwise from north, Broom-More, Tavnaghboy, Kilrobert, Clare Mountain, Aghaleck, Corvally, Essan, Cleggan, Stroan, and Tullaghore.
It sounds like you are in the U.K. If you could locate such a map (you can send it to me privately) it might be worth adding to my paper.
While not exactly related to the math content, I was interested in the Siciliy map, partly because I've always been fascinated by the four color theorem but also becasue I was recently there. Last summer, I visited Siciliy with friends, and made a point of climbing to the summit while I was there. As we sat at an outside bar in the evening with a view of the mountain, it erupted. It has been much more active this year, but I caught it just as it was starting to get more active. I'm sorry to say I hadn't appreciated the map implications at the time, but I do now. Thanks.
Wow! Did you get any good pictures of the eruption?
Not that it is not true that the number of colors is bounded. This is a big open question. The main conjecture is that if ≤q countries are allowed to share a point, then the number of colors needed is at most 3/2 q. This is true when q=3 by the Four Color Theorem. And Borodin proved it when q=4. But it is not proved when q=5. Oh: It has been proved also when q=6.
I omitted the word “planar” when quoting your theorem. Is that the mistake?
In your paper, you quote a result by Ore and Plummer that says that the chromatic number of the facial closure is bounded above by 2q. Doesn’t that prove that the number of colours is “bounded as a function of q”? I thought that the big open question was that the optimal bound is 3q/2.
You are right about the 2q bound. I thought you were referring to 3/2 q, which you were not. So there is no error, except that, yes, planar must be mentioned. Note that the dual of a map is generally called a graph. But it is not wrong to call it a map. We have just submitted our work for publication, so we will see what editor/referees say. It was exciting to find the Sicilian anomaly.
Thanks for the clarification. I have added the word “planar”.
I used the words “map” and “region” in some places instead of “graph” and “face” in an effort to keep the post relatable for a general audience.
There are 10 towns in N Ireland that meet at a point. See the wikipedia entry on "Quadripoints" where one finds: In Northern Ireland, ten townlands meet at the summit of Knocklayd.[62][63][64] The townlands are, clockwise from north, Broom-More, Tavnaghboy, Kilrobert, Clare Mountain, Aghaleck, Corvally, Essan, Cleggan, Stroan, and Tullaghore.
It sounds like you are in the U.K. If you could locate such a map (you can send it to me privately) it might be worth adding to my paper.
Can you confirm: Is it (was it?) possible to get to the common point to the 10 towns? Stan Wagon, author of paper.