A Practical 4-coloring Method of Planar Graphs Mingshen Wu1 and Weihu Hong2 1Department of Math, Stat, and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751 2Department of Mathematics, Clayton State University, Morrow, GA, 30260 ABSTRACT An interesting coloring formula was conjectured by The existence of a 4-coloring for every finite loopless planar Heawood in 1890 [7]: For a given genus g > 0, the minimum graph (LPG) has been proven even though there are still number of colors necessary to color all graphs drawn on the some remaining discussions on the proof(s). How to design surface of that genus is given by a proper 4-coloring for a LPG? The purpose of this note is 7++ 1 48g to introduce a practical 4-coloring method that may provide γ ()g = . 2 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 For example, with genus g =1, it is a doughnut-shaped coloring. object. γ (1)= 7 means the chromatic number is seven. Picture 2 shows that the seven regions (with seven different Key words: 4-coloring, two-color subgraph, triangulation colors) can be topologically put on a torus such that the seven regions are mutually adjacent each other on the torus. 1 Brief historical review So, it clearly requires at least seven colors to be able to give In 1852, Francis Guthrie, a college student, noticed that only a proper coloring of maps on the torus. 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 working 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. 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 known as face coloring problem. We Picture 3: the Klein bottle [9] denote a finite loopless (every edge joins two distinct vertices) planar (can be drawn on a plane without edge A plane map can be seen as a map on a globe (picture 4), and crossing) graph by LPG. The dual graph D(G) of a LPG G is hence 4-colring of planar graphs is equivalent to 4-coloring defined as the graph that represents each face of G by a of the maps on a globe. A torus with g=0 is isomorphic to a vertex, and two vertices are adjacent in D(G) if and only if globe. So, even if the proof of 4-coloring theorem was the the two faces share boarder in G, for instance, the dotted most difficult one, we may see the 4-coloring of maps is the graph in 1(b) is the dual of 1(a). Clearly, D(G) is a LPG as lowest case of Heawood conjecture. 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 Haken proved the 4-coloring theorem in 1976 using a computer [1],[2],[3]. However, many mathematicians have been working on a theoretical proof [4], some individuals tried to give a counter proof [5]. Some mathematicians have Picture 4: A globe to a plane been studying the properties and invariants of the 4-coloring of LPGs as well [6]. The 4-coloring problem is still drawing attention around the world. People also would like to see an effective method that can provide a 4-coloring for a map or a LPG that people subgraphs: G1,2 and G3,4 , G1,3 and G2,4 , and G1,4 and G2,3 . are interested in. For example, picture 6(a) shows a proper 4-coloring of the Heawood graph. Picture 6(b) shows a pair of two-color 2 Some definitions related to a LPG subgraphs. 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. 6(a) 6(b) Picture 6: the Heawood graph 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 Picture 5: a map, its dual graph and triangulation tree structure, i.e., each component is a cycle, or a tree, or a “cycle-tree” (tree is hanging on the cycle). For example, Mathematicians have studied the properties of such picture 6(b) shows a cycle and a tree for the two black color maximum planar graphs. The famous Euler formula says symbols; a tree and a cycle-tree for the two white color symbols. Each has only even cycle(s), if any, for a that, let p , e , and f be the number of vertices, edges, and Gij, faces of a planar graph G, respectively, then pe−+ f =2. proper 4-coloring. So, trying to break any two-color odd cycle of an improper 4-coloring is a critical scheme for If G is also a triangulation, then one can derive the following getting a proper one. properties from Euler formula: (a) = − , ep36 A fact that can help on finding a proper 4-coloring is that if a (b) fp=2 − 4, and vertex is of degree three or lower, we may remove it from (c) 6pe−= 2 12 . [(c) shows that in a triangulation planar the graph since the color of this vertex is either uniquely graph, there is at least a vertex with degree ≤ 5.] determined by its neighbors or it can be an adjustable vertex when we put them back. This action can be performed Definitions: Suppose a planar graph G receives a 4-coloring recursively before design a coloring. using colors c , c , c ,and c . Assume that vertex v is 1 2 3 4 We define a reference path P of a graph G to be a path of G properly colored referring to its neighborhood. We say that such that no subset of vertices of P forms an odd cycle in G. vertex v is adjustable if the color of v can be assigned a A reference path can be properly colored by two colors that different color, without changing any color of its neighbors, are alternately applied along the path. such that v itself is still properly colored referring to its neighbors. For example, the vertex v in picture 6 is 3 The practical method of finding a 4-colring 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 of a LPG improper coloring on a bad-edge via color switching This note introduces a programmable process that may schemes. 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. Picture 6: an adjustable vertex Step 1: Find and remove all vertices with degree three or lower recursively. [Stop if the remaining number of vertices Assume that G receives a proper 4-coloring. A two-color is less than 5. A proper 4-coloring is obvious.] subgraph is a ccij- subgraph Gij, spanned by all vertices of Step 2: select a reference path P and color it by two colors, G that are colored by ci or c j . There are six two-color say c and c , alternately. [User should be able to select the subgraphs. We consider them as three pairs of two-color 1 2 reference path via user-interface.] (4.3.3) Detect whether there is an adjustable vertex. If so, Step 3: (Initial coloring) assign a different color to the adjustable vertex and return to Let Sk be the ordered set of all vertices v of G such that the beginning of step 4. Otherwise, continue. distance DvP(, )= k in G. The order of vertices in S is k Step 5: Assign a new reference path P and go to step 3 to created by a depth first search. For example, if x is the last restart the coloring process. The reference path P can have vertex added to Sk , then a neighbor y of x with DyP(, )= k, one or more vertices. User may select P by experience or observation. if any, should be added to Sk next. So, Sk is ordered like a circle around the reference path P. 4 Examples As you have seen above, we represent the four colors using Color by colors c and c if k is odd, or by c and c if k Sk 3 4 1 2 the symbols ▲, , ∆, and O. is even.