The Four Color Problem What is the minimum number of colors needed to color any map so that adjacent regions do not have the same color? Lucia Maddalena Priulla Index: - History of the Four-Color Problem; - Kempe’s approach: the first proof of the Four-Color Theorem; - Heawood counterexample to Kempe’s proof; - The conjectures of Hajòs and Hadwiger; - The Four-Color Theorem as a corollary of Hadwiger’s Conjectures; - More result about coloring: Mycielski’s Construction; - Mader’s Theorem; - König’s Theorem; - Vizing’s Theorem; - Bibliography. Some mathematicians attribute the problem to August Ferdinand Möebius. It is said that during a lecture, the lecturer turn to his students the following problem: “There was a king with five sons. In his will, he stated that after his death his kingdom should be divided into five regions in such a way that each region should have a common boundary with the other four. Can the terms of the will be satisfied?” • Möebius said that this problem cannot be solved. • In 1878 Alfred Cayley, introduced this problem to London Mathematical Society. The Four Color Problem •In 1852 Francis Guthrie (1831-1899), a former graduate of University College London, observed that the countries of England could be coloured with 4 colours so that neighboring countries are coloured differently. This, led him to ask if the previous statement holds for every map (real or imagined). •Francis mentioned this problem to his brother Frederick, who was taking a class from the famous mathematicians August De Morgan. •De Morgan was unable to solve this problem, and other mathematicians seemed to share this interest. •In 1879 there was an article written by the British lawyer Alfred Bray Kempe containing a proposed proof that every map can be coloured with 4 or fewer colours so that neighboring countries are coloured differently. •For the next ten years, the Four Color Problem was considered to be solved. •An 1890 article by the British mathematician Percy John Heawood showed a counterexample to the technique used by Kempe; so it showed that Kempe’s method was unsuccessful. The idea of Kempe. Now, we show the steps of the Kempe’s proof, which it contains an error. To see this, we need to introduce some new definitions. Recall Definition: A graph is planar if it has a drawing without crossing. A plane graph is a drawing of a planar graph in the plan. Definition: a map is a planar graph G, whose faces have sides which are simple and closed curve, called regions, and it satisfies: 1) Every side divides different faces; 2) Every vertex v has degree greter or equal to 3. Every map is a planar graph, but it is not true the vice versa. KEMPE CHAIN ARGUMENT Kempe chain argument in vertex colouring: Let G be a graph with a colouring using at least two different colours represented by i and j. For example i=1, and j=2. Let H(i, j) denote the subgraph of G induced by all the vertices of G coloured either i (=1) or j (=2) and let K be a connected component of the subgraph H(i, j). If we interchange the colours 1 and 2 on the vertices of K and keep the colours of all other vertices of G unchanged, then we get a new colouring of G, which uses the same colours with which we started. This subgraph K is called a Kempe chain and the recolouring technique is called the Kempe chain argument. Edge-colourings Definition: A k-edge-colouring of G is a labeling f : E(G) --> [k]; the labels are “colours”. A proper k-edge-colouring is a k-edge-colouring such that edges sharing a vertex receive different colours; equivalently, each colour class is a matching. A graph G is k-edge-colourable if it has a proper k-edge-coloring. The edge-chromatic number or chromatic index χ’(G) is the minimum k such that G is k-edge colourable. Kempe chain argument for edge colouring: it is the same topic of Kempe chain argument for vertex colouring. It’s enough to replace vertex with edge. Definition: consider a map colored with four colors: a,b,c,d. A Kempe’s chain (or ac-chain) is a sequence of regions (faces) colored with colors a and c, without passing from regions colored with b or d. ac-chain Kempe’s approach for solving the Four Color Problem. He considered a region R in a map M such that R is surrounded by five regions and showed that for every coloring of M with four colors, there is a coloring of the entire map M with four colors. If only three colors are used to color these five regions, then a color is available for R, so we have obtained a 4-coloring. We consider a map with 4 colors: yellow, green, blue, red. Only the region R1 is colored yellow. Consider all the regions of the map M that are colored either yellow or red and that, beginning at R1, can be reached by an alternating sequence of neighboring yellow and red regions, that is, by a yellow-red Kempe chain. If the region R3 (which is the neighboring region of R colored red) cannot be reached by a yellow-red Kempe chain, then the colors yellow and red can be interchanged for all regions in M that can be reached by a yellow-red Kempe chain beginning at R1. This results in a coloring of all regions in M (except R) in which neighboring regions are colored differently and such that each neighboring region of R is colored red, blue, or green. We can then color R yellow to arrive at a 4-coloring of the entire map M. From this, we may assume that the region R3 can be reached by a yellow-red Kempe chain beginning at R1. See the figure : Let’s now look at the region R5, which is colored green. We consider all regions of M colored green or red and that, beginning at R5, can be reached by a green-red Kempe chain. If the region R3 cannot be reached by a green-red Kempe chain that begins at R5, then the colors green and red can be interchanged for all regions in M that can be reached by a green-red Kempe chain beginning at R5. Upon doing this, a 4-coloring of all regions in M (except R) is obtained, in which each neighboring region of R is colored red, blue, or yellow. We can then color R green to produce a 4-coloring of the entire map M. We may therefore assume that R3 can be reached by a green-red Kempe chain that begins at R5. (See Figure.) Because there is a ring of regions consisting of R and a green-red Kempe chain, there cannot be a blue-yellow Kempe chain in M beginning at R4 and ending at R1. In addition, because there is a ring of regions consisting of R and a yellow-red Kempe chain, there is no blue-green Kempe chain in M beginning at R2 and ending at R5. Hence we interchange the colors blue and yellow for all regions in M that can be reached by a blue-yellow Kempe chain beginning at R4 and interchange the colors blue and green for all regions in M that can be reached by a blue-green Kempe chain beginning at R2. Once these two color interchanges have been performed, each of the five neighboring regions of R is colored red, yellow, or green. Then R can be colored blue and a 4-coloring of the map M has been obtained, completing the proof. Heawood counterexample to Kempe’s proof. Blue-yellow-chain and blue-green-chain Kempe’s proof is unsuccessful when it is applied to the Heawood map, because there are neighboring regions with the same color. So, Heawood had discovered a counterexample to Kempe’s technique, not to the Four Neither red-yellow-chain nor red- Color Conjecture green-chain are present. itself. •Nevertheless,Heawood was able to use Kempe’s technique to prove that every map could be colored with five or fewer colors. •The Four Color Problem can be stated strictly in terms of planar graphs, rather then in terms of maps. Definition: Let G be a graph. Then G is k-region-colorable if each region of G can be assigned one of k given colors so that neighboring (adjacent) regions are colored differently. The Four Color Conjecture: Every plane graph is 4-region-colorable. There is another more popular statement of the Four Color Conjecture which involves the coloring of vertices. Definition: Let G be a graph. The planar dual G* of G can be constructed by first placing a vertex in each region of G. This set of vertices is V(G*). Two distinct vertices are then joined by an edge for each edge on the boundaries of the regions corresponding to these vertices of G*. Furthemore, a loop is added at a vertex of G* for each bridge of G on the boundary of the corresponding region. Each edge of G* is drawn so that it crosses its associated edge of G, but crosses no other edge of G or G*. Thus, G* is planar. Since G* may contain parallel edges and possibly loops, G* is a multigraph. The dual G* has the properties that its order is the same as the number of regions of G, and the number of regions of G* is the order of G.