A Practical 4-coloring Method of Planar Graphs

Transcription

1 A Practical 4-coloring Method of Planar Graphs Mingshen Wu 1 and Weihu Hong 2 1 Department of Math, Stat, and Computer Science, University of Wisconsin-Stout, Menomonie, WI Department of Mathematics, Clayton State University, Morrow, GA, ABSTRACT The existence of a 4-coloring for every finite loopless planar graph (LPG) has been proven even though there are still some remaining discussions on the proof(s). How to design a proper 4-coloring for a LPG? The purpose of this note is to introduce a practical 4-coloring method that may provide a proper 4-coloring for a map or a LPG that users may be interested in. The coloring process may be programmable with an interactive user-interface to get a fast proper coloring. Key words: 4-coloring, two-color subgraph, triangulation 1 Brief historical review In 1852, Francis Guthrie, a college student, noticed that only four different colors were needed while he was trying to color the map of counties of England such that no two adjacent counties receive a same color. Since then, mathematicians had been woring hard to prove this seemingly easy but full of traps problem. Clearly, four colors are definitely needed even for a simple map as shown by picture 1(a) below. An interesting coloring formula was conjectured by Heawood in 1890 [7]: For a given genus g > 0, the minimum number of colors necessary to color all graphs drawn on the surface of that genus is given by g γ ( g) =. 2 For example, with genus g = 1, it is a doughnut-shaped object. γ (1) = 7 means the chromatic number is seven. Picture 2 shows that the seven regions (with seven different colors) can be topologically put on a torus such that the seven regions are mutually adjacent each other on the torus. So, it clearly requires at least seven colors to be able to give a proper coloring of maps on the torus. Picture 2: Seven coloring torus [9] Ringel and Youngs [8] proved that Heawood conjecture is true except the Klein bottle which needs six colors only (picture 3). 1(a) 1(b) Picture 1: face coloring is equivalent to vertex coloring Coloring a map is nown as face coloring problem. We denote a finite loopless (every edge joins two distinct vertices) planar (can be drawn on a plane without edge crossing) graph by LPG. The dual graph D(G) of a LPG G is defined as the graph that represents each face of G by a vertex, and two vertices are adjacent in D(G) if and only if the two faces share boarder in G, for instance, the dotted graph in 1(b) is the dual of 1(a). Clearly, D(G) is a LPG as well. A LPG is 4-colorable if its dual graph is vertex 4- colorable. So, face coloring is studied via vertex coloring. Two American professors Kenneth Appel and Wolfgang Haen proved the 4-coloring theorem in 1976 using a computer [1],[2],[3]. However, many mathematicians have been woring on a theoretical proof [4], some individuals tried to give a counter proof [5]. Some mathematicians have been studying the properties and invariants of the 4-coloring of LPGs as well [6]. Picture 3: the Klein bottle [9] A plane map can be seen as a map on a globe (picture 4), and hence 4-colring of planar graphs is equivalent to 4-coloring of the maps on a globe. A torus with g=0 is isomorphic to a globe. So, even if the proof of 4-coloring theorem was the most difficult one, we may see the 4-coloring of maps is the lowest case of Heawood conjecture. Picture 4: A globe to a plane The 4-coloring problem is still drawing attention around the world. People also would lie to see an effective method

2 that can provide a 4-coloring for a map or a LPG that people are interested in. 2 Some definitions related to a LPG A triangulation of a LPG G may be obtained by adding edges, without edge crossing, to G until each face of G becomes a triangle (picture 5). Such a triangulation is also said to be a maximum planar graph. Obviously, a LPG is definitely 4-colorable if its triangulation is 4-colorable. subgraphs: G and G 1,2 3,4, G and 1,3 G 2,4, and G 1,4 and G 2,3. For example, picture 6(a) shows a proper 4-coloring of the Heawood graph. Picture 6(b) shows a pair of two-color subgraphs. 6(a) 6(b) Picture 6: the Heawood graph Picture 5: a map, its dual graph and triangulation Mathematicians have studied the properties of such maximum planar graphs. The famous Euler formula says that, let p, e, and f be the number of vertices, edges, and faces of a planar graph G, respectively, then p e+ f = 2. If G is also a triangulation, then one can derive the following properties from Euler formula: (a) e= 3p 6, (b) f = 2 p 4, and (c) 6 p 2e= 12. [(c) shows that in a triangulation planar graph, there is at least a vertex with degree 5.] Definitions: Suppose a planar graph G receives a 4-coloring using colors c 1, c 2, c 3,and c 4. Assume that vertex v is properly colored referring to its neighborhood. We say that vertex v is adjustable if the color of v can be assigned a different color, without changing any color of its neighbors, such that v itself is still properly colored referring to its neighbors. For example, the vertex v in picture 6 is adjustable. An edge is said to be a bad-edge of a coloring if its two endpoints received the same color. We will fix the improper coloring on a bad-edge via color switching schemes. Picture 6: an adjustable vertex Assume that G receives a proper 4-coloring. A two-color subgraph is a c i -c j subgraph G spanned by all vertices of i, j G that are colored by c i or c j. There are six two-color subgraphs. We consider them as three pairs of two-color In a two-color subgraph, the total degree of each component must be even, and hence the total degree of a two-color subgraph is even. These two-color subgraphs also have the following properties: Each component of a two color subgraph is of cycle and/or tree structure, i.e., each component is a cycle, or a tree, or a cycle-tree (tree is hanging on the cycle). For example, picture 6(b) shows a cycle and a tree for the two blac color symbols; a tree and a cycle-tree for the two white color symbols. Each G has only even cycle(s), if any, for a i, j proper 4-coloring. So, trying to brea any two-color odd cycle of an improper 4-coloring is a critical scheme for getting a proper one. A fact that can help on finding a proper 4-coloring is that if a vertex is of degree three or lower, we may remove it from the graph since the color of this vertex is either uniquely determined by its neighbors or it can be an adjustable vertex when we put them bac. This action can be performed recursively before design a coloring. We define a reference path P of a graph G to be a path of G such that no subset of vertices of P forms an odd cycle in G. A reference path can be properly colored by two colors that are alternately applied along the path. 3 The practical method of finding a 4-colring of a LPG This note introduces a programmable process that may provide a proper 4-coloring for a general map or a LPG. We do not assert this process must be a success (that would claim we have a proof of the 4-coloring theorem), however, we have successfully found a proper 4-coloring for all examples we have had via this method. Assume that we are given a planer graph G. Step 1: Find and remove all vertices with degree three or lower recursively. [Stop if the remaining number of vertices is less than 5. A proper 4-coloring is obvious.] Step 2: select a reference path P and color it by two colors, say c 1 and c 2, alternately. [User should be able to select the reference path via user-interface.]

3 Step 3: (Initial coloring) Let S be the ordered set of all vertices v of G such that distance DvP (, ) = in G. The order of vertices in S is created by a depth first search. For example, if x is the last vertex added to S, then a neighbor y of x with DyP (, ) =, if any, should be added to S next. So, S is ordered lie a circle around the reference path P. Color S by colors c 3 and c 4 if is odd, or by c 1 and c 2 if is even. This step gives an initial coloring for all vertices of G. [There may have bad-edge(s) that will occur if either any S itself forms an odd cycle or a subset of S forms an odd cycle in G.] Step 4: (Correction step) Detect any bad-edge(s). If no bad-edge, it is a perfect coloring and stop. Otherwise, fix the bad-edges one at a time (of course, if lucy, more than one bad-edge might be fixed simultaneously). Select a bad-edge (u, v) and assume u and v received color c 1. (4.1) For each two-color subgraph G, j=2, 3, or 4, detect 1, j whether there is a component that consists of only even cycle(s) and/or tree(s), and that contains exactly one of u and v. If so, the bad-edge may be fixed by swapping these two colors on this component, and return to the beginning of step 4. Otherwise, continue. (4.3.3) Detect whether there is an adjustable vertex. If so, assign a different color to the adjustable vertex and return to the beginning of step 4. Otherwise, continue. Step 5: Assign a new reference path P and go to step 3 to restart the coloring process. The reference path P can have one or more vertices. User may select P by experience or observation. 4 Examples As you have seen above, we represent the four colors using the symbols,,, and O. Example 1: The Heawood graph (picture 6(a)) (see [10]). Heawood presented this graph as a counterexample to the proof of 4-coloring theorem by Alfred Kempe. 4-coloring process by the practical method: Step 1: Nothing can be done. Step 2: Select the circled two vertices as the reference path and color it using the two blac color symbols. (7(a)) Step 3: Color the ordered set S alternately using two white color symbols if is odd; two blac color symbols if is even. There is a bad-edge (u, v) with color in S 5 (7(a)). Step 4: (4.1) No such component. (4.2) There is an odd cycle within each of (, O), (, ) and (, ) subgraphs as shown on picture 7(b). This implies we have no way to fix (u, v) now. (4.2) For each two-color subgraph G, j=2, 3, or 4, detect 1, j whether there is a component that consists of two-color even cycle(s) and/or tree(s) containing edge (u, v). If so, recolor this component with the two colors, and return to the beginning of step 4. (u, v) should be fixed. Otherwise, there is an odd cycle containing (u, v) in each two-color subgraph. (u, v) cannot be fixed now. Picture 7(a) Picture 7(b) (4.3) This step does not fix any bad-edge, but change the color status. Users should have a chance to determine a starting vertex to perform any one of the following sub-steps. (4.3.1) Detect whether there is a component C in a two-color subgraph with one color of S and one color of S + such 1 that C consists of even cycle(s) and/or tree(s) only. If such a component exists, exchanging the two colors along C, then return to the beginning of step 4. If necessary, this detection can be performed several times using different starting vertex, and/or different pair of colors between S and S +. Continue if nothing can be done. 1 (4.3.2) Detect whether there is a two-color cycle that contains edge (u, v) such that we can change the color c 1 that u and v both have now to another color. [This does not create or fix a bad-edge.] If yes, do so and return to the beginning of step 4. Otherwise, continue. Picture 7(c) Picture 7(e) Picture 7 Picture 7(d) Picture 7(f)

4 (4.3) starting with vertex x picture 7(c) (4.3.1) There is a -O tree cross S s from vertex x (7(c)). Swap color and O along this tree, and return to the beginning of step 4. (7(d)) (4.1) There is a - component that is a tree containing vertex u only (indicated by the broen-curve in picture 7(e)). Swap colors and along this component and return to the beginning of step 4. No more bad-edge. A perfect coloring obtained. (7(f)) Step 1: Done already. Step 2: Select the two vertices at the center of the graph as the reference path. Color this path by and. (picture 11) Step 3: Search for S, for =1 to 9. Color each S using two white color symbols if is odd; or two blac color symbols if is even. S is of only three vertices. Unfortunately, 9 there are odd cycles within S and 8 S. (picture 11) 9 Note: This coloring process is not unique. Referring to picture 7(d), v by itself is a trivial - path, so we can simply switch color to for v to get a perfect 4-coloring as well. Example 2: Gardner s April Fools Day puzzle In 1975, Martin Gardner claimed the map of 110 regions (picture 8) requires five colors to get a proper coloring. Of course, Gardner just made a fun on the Fool s Day. A proper 4-coloring of Gardner s map had been found couple years after he published the map (picture 9). Picture 11 After coloring by distance to the reference path P, there are several bad-edges. See the broen lines in picture 12. Step 4: Select a bad-edge (u, v) with color (picture 12). Picture 8 Picture 9 We illustrate the coloring process by find a proper 4-coloring of the Gardner s graph. Since region 110 (see picture 8) is enclosed by three regions only, so its color will be uniquely determined by the surrounding three regions and hence we can remove it from the map. The dual triangulation of Gardner s map (with region 110 removed) is given by picture 10. We too this picture from Xu s boo [6] and corrected a minor error. Picture 12 (4.1) Starting with vertex u there is a - path that contains vertex u only. Swap and and return to the beginning of step 4. [Lucily, three bad-edges on the left side are all fixed. Picture 13] Picture 10 Gardner graph

5 (4.3.2) There is an even -O cycle found (picture 14). Exchange color and O along this cycle and return to the beginning of step 4. [Now, the color of both u and v is (picture 15).] (4.1) Starting with vertex u, there is a - component that consists of even cycles and trees and that does not contain vertex v (picture 15). Swapping colors and for this component results in a perfect coloring of Gardner s graph as shown in picture 16. Picture 13 Step 4: The only bad-edge (u, v) is on the right side which received color O for both u and v (picture 13). (4.1) No such component in anyone of two-color subgraphs. (4.2) There is an odd-cycle found in each (O, ), (O, ), or (O, ) subgraphs. This implies that we have no way to fix the bad-edge by switching a pair of colors now. Picture 13 shows the three two-color odd cycles. Under this situation our process goes to (4.3). Hope (4.3) can update the color situation and possible have a chance to get rid of the bad-edge. Picture 16 A perfect 4-coloring Example 3: Let s study the Triais Icosahedral graph [11] (picture 17(a)). To directly design a proper 4-coloring for this graph is not trivial since there are many K s. However, removing 4 all vertices with degree three (step 1 of our coloring procedure) gave graph 17(b). We omit the details since it is not difficult at all to assign a proper 4-coloring for 17(b) as shown in 17(c). A proper 4-coloring obtained by adding those removed vertices bac with a proper color (see 17(d)). Picture 14 17(a) 17(b) Picture 15 17(c) Picture 17 17(d)

6 5 Conclusions The 4-coloring process we introduced here can be done manually. It is an interesting exercise that can be a practical project for students. This process is also programmable. In fact, the fundamental detection process includes DFS or BFS search, creating the ordered sets, assigning colors, and swapping colors actions. User-interface is an important factor for getting a fast result. [10] Wolfram MathWord, ColorGraph.html [11] Wolfram MathWord, /TriaisIcosahedralGraph.html We finish this note by indicating that an actual map might requires more than four colors if there are some countries whose territory has more than one region and the territory of each country must be of the same color. For example, in the following map, if the two regions both labeled 1 must be of the same color, it would enforce to use the fifth color. 6 REFERENCES [1] K. Appel and W. Haen, Every planar map is four colorable. Part I. Discharging, Illinois J. Math. 21 (1977), 429{490. MR 58:27598d [2] K. Appel, W. Haen, and J. Koch, Every planar map is four colorable. Part II. Reducibility, Illinois J. Math 21 (1977), 491{567. MR 58:27598d [3] K. Appel and W. Haen, Every planar map is four colorable, A.M.S. Contemporary Math. 98 (1989). MR 91m:05079 [4] Electronic Research Announcements of the American Mathematical Society,. Volume 2, Number 1, August 1996 [5] [6] Shouchun Xu, 图说四色问题 (Picturizing 4-coloring problem), Peing University Press, ISBN , [7] Heawood, Map colour theorem, Quart. J. Pure Appl. Math. 24 (1890) [8] Ringel and Youngs, Solution of the Heawood mapcoloring problem, Proc. Nat. Acad. Sci. USA 60 (1968) [9] Wiipedia, File: Projection_color_torus.png