THE FOUR-COLOR THEOREM. Joshua D. Chesler

Transcription

1 THE FOUR-COLOR THEOREM by Joshua D. Chesler A Thesis Submitted to the Faculty of the DEPARTMENT OF MATHEMATICS In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE In the Graduate College THE UNIVERSITY OF ARIZONA

2 2 TABLE OF CONTENTS LIST OF FIGURES ABSTRACT CHAPTER 1 Introduction Graph Theory History Kempe Chains Map-Coloring in Higher-Genus Spaces The Five-Color Theorem CHAPTER 2 The Four-Color Theorem CT for Maps and Graphs Duals of Graphs More Equivalent Conditions Overview of the Proof by Robertson et al CHAPTER 3 Configurations Triangulations Minimal Counterexamples More on Kempe Chains Configurations Good Configurations Tricoloring Contraction Tri-coloring of G Modulo X CHAPTER 4 Reducibility Consistent Sets of Edge Colorings Free Completion D-reducibility Contracts Projections Triads Proof of Theorem CHAPTER 5 Unavoidability Cartwheels Passes Rules P satisfies Theorem Discharging Vertices

3 3 TABLE OF CONTENTS Continued CHAPTER 6 An Example CHAPTER 7 A Coloring Algorithm CHAPTER 8 Conclusion REFERENCES

4 4 LIST OF FIGURES 1.1 Two drawings of complete graph on 4 vertices, K 4, a planar graph. The drawing on the right is a plane drawing, the drawing on the left is not. An example of a non-planar graph is given in Figure From left to right: We have a graph G with edge e, G with edge e deleted, and G with edge e contracted The complete graph on 5 vertices, K 5, is not planar and is not four-colorable A triangulation. Note this graph is 4-colorable but not 3-colorable Permissible circuits of size 5 with interior edges A short circuit with interior edges & vertex Examples of configurations Graph G is a subgraph of G but it is not an induced subgraph A near-triangulation G with no short circuit A circuit R A free completion of Figure 4.3 with ring-size A Configuration K. Recall, vertex shape γ K (v) = 6, γ K (v) = A good configuration with thickened half-edges represented by double edges A free completion of the leftmost configuration in Figure 3.4. Vertex v is a triad for X = {x 1, x 2, x 3, x 4 } A free completion of Figure 4.4. Vertex v is a triad for the thickened/doubled edges Example of a cartwheel A subgraph of T Examples of rules Pass P obeys the left-most rule in Figure 5.3 and appears in cartwheel W with hub s An internally 6-connected triangulation T The components v, C 1, and C 2 of the cartwheel W A good configuration K appearing in W The Birkhoff diamond S is the free-completion of K

5 5 ABSTRACT The four color theorem states that every loopless planar graph admits a vertex fourcoloring. This was proved in 1976 by Appel and Haken; the proof was improved upon in the 1990s by Robertson et al. Both versions of the proof relied upon a computer. Here we explain the proof of Robertson et al; we focus on the pieces which lead up to and justify the use of a computer.

6 6 CHAPTER 1 Introduction The Four-Color Theorem (4CT) states that every map can be colored with no more than four colors. More specifically, we require that any two regions (states, countries, etc.) that share a border of any length cannot be colored the same color. Thus, for example, Colorado and Arizona could be colored the same color as their borders intersect in only a point. We require that our regions are contiguous; thus countries like the United States with unattached territories like Alaska and Hawaii will not be considered. Given these few conditions, one would think that the more complex the map the more colors required. However, for any map on the plane or sphere four colors suffice. This famous theorem was finally proved in the 1970 s after 120+ years of notoriety. The Four-Color Conjecture attracted the attention of many famous mathematicians and even was thought to have been proved for ten years in the 19th century. The proof finally offered in the 70 s by Appel and Haken (A&H) (Appel and Haken, 1977; Appel et al., 1977) was aided by a computer. Though the proof has been improved upon it still is dependent upon computer assistance. Herein, we explore the most recent proof by Robertson et al (Robertson et al., 1997). 1.1 Graph Theory The Four-Color Theorem can be formulated in the language of Graph Theory. A graph G is a finite set of vertices V(G) and a set of edges E(G) joining the vertices. The degree of a vertex v in a graph G, denoted d G (v), is the number of edges that has that vertex as

7 an end-point. We say a graph has multiple edges if two vertices are joined by more than one edge. An edge which has the same vertex as both it s end-points is called a loop (a loop contributes 2 to the degree of a vertex). A simple graph has no loops or multiple edges. A graph in which every vertex has the same degree is a regular graph. A complete graph is one in which every pair of vertices is joined by exactly one edge. Two vertices v, w V(G) are adjacent if there is an edge vw joining them; in this case, we say vertices v and w are incident with edge vw. A graph G is a subgraph of graph G if V(G ) V(G) and E(G ) E(G). Two graphs G 1 and G 2 are isomorphic if there is a bijection between V(G 1 ) and V(G 2 ) that preserves adjacency and incidence relationships. A planar graph is one which is isomorphic to a graph in the plane without any crossings; that is, edges only intersect at vertices. A drawing of such a graph without crossings is called a plane graph. If G is a planar graph then it divides the plane into regions, called faces. We will justify restating the Four-Color Theorem as a statement about simple planar graphs. Figure 1.1: Two drawings of complete graph on 4 vertices, K 4, a planar graph. The drawing on the right is a plane drawing, the drawing on the left is not. An example of a non-planar graph is given in Figure A sequence of adjacent edges in a graph is called a walk. If no vertex is repeated then it is called a path. A graph is connected if any two vertices are connected by a path. A closed path (i.e., starts and ends with the same edge) is called a circuit; equivalently, a circuit is a non-null connected graph in which every vertex is degree 2. A tree is a connected graph with no circuits; a forest is the union of trees. We can obtain subgraphs of a graph G by deleting vertices and edges. If e E(G)

8 then G \ {e} is the graph obtained by deleting edge e. If v V(G) then G \ {v} is the graph obtained by deleting vertex v and all edges incident with v. We may thus define how connected a graph is. A graph G is k-connected if at least k vertices must be deleted to disconnect the graph. We similarly define k-edge connected for edges. We may also contract an edge e by removing it and identifying its end-vertices v and w so that the new vertex is incident with all edges (other than e) which were originally incident with v or w. e Figure 1.2: From left to right: We have a graph G with edge e, G with edge e deleted, and G with edge e contracted. A (vertex) k-coloring for graph G is a map c : V(G) {1,..., k} such that c(u) c(v) edges in E(G) with ends u and v. We will show that the Four-Color Theorem is equivalent to the statement that simple planar graphs admit a vertex four-coloring History Francis Guthrie, a former student of Augustus de Morgan, is credited with first proposing the Four-Color Conjecture in He made the observation while coloring a map of England; the problem soon made it s way to de Morgan who, fascinated by the problem, immediately wrote of it to Hamilton. Hamilton did not share de Morgan s interest in the problem. However, in the decades to come, it attracted the attention of many prominent mathematicians. Charles Peirce and Arthur Cayley both investigated the conjecture in the 1870 s and in 1879 Alfred Bay Kempe, a former student of Cayley s, published a proof of the theorem. His proof employed an argument known as Kempe Chains, described below (Section 1.2.1). Kempe s proof was accepted by the mathematical community until 1890 when Percy John Heawood published a paper showing a defect in the proof. Another proof offered by P.G. Tait in 1880 was found to be faulty in The Four-Color Theorem

9 9 once again became The Four-Color Conjecture for nearly 100 more years. Heawood s interest in map coloring continued throughout his life; he made many contributions on the subject including a proof of the Five Color Theorem (Theorem 1.4.3). The conjecture continued to hold the interest of mathematicians throughout the 20th century. In 1976, Appel & Haken offered a new and valid proof for the 4CT (Appel and Haken, 1977; Appel et al., 1977). It built upon the idea of reducibility developed by George David Birkhoff and used many of the tools of their predecessors, such as Kempe Chains and Heinrich Heesch s idea of an unavoidable set of configurations. Appel & Haken used a novel approach to complete their proof; after improving upon Heesch s ideas, they finished the proof with a computer. After 1200 hours of computer time, the 4CT became the first major theorem to be proved with the aid of a computer. More recent work by Robertson et al (Robertson et al., 1997) has led to improvements upon Appel & Haken s proof. Their set of configurations consisted of 633 elements, rather than the 1400 of A&H. They use 32 rules for proving unavoidability (versus 487). Thus, though their proof still relies upon a computer for checking reducibility, it does so to a lesser extent. Further, their proof is machine checkable using logic-checking programs such as Coq (Knight, 2005) Kempe Chains Despite the flaws in Kempe s proof of the 4CT, the methods of his proof remain vital to the proof presented herein. In particular, Kempe used a method known as Kempe chains. Suppose M is a map in which every region (country) is colored red, green, yellow, or blue except for one region X. If the regions bordering X are only colored with three of these colors then we may color X with the fourth color to obtain a four-coloring. If the regions bordering X are colored with all four colors then there are some additional complications. Consider four regions A, B, C, D which border X and are colored red, green, yellow, blue respectively. Then there are two cases:

10 10 1. There is no chain of adjacent regions from A to C alternately colored red and yellow. 2. There is a chain of adjacent regions from A to C alternately colored red and yellow. If (1) holds then we may interchange colors in the chain. Thus, A is colored yellow. We may reiterate until there are no more red regions bordering X. Then X may be colored red. If (2) holds, then there can be no chain of green and blue regions from B to D (else it would cross the red-yellow chain) and a similar argument will allow us to color X. Kempe s attempt at a proof depended heavily on this method of recoloring. It was more than ten years before Heawood found a flaw in his logic. Kempe s proof relied upon simultaneous recolorings. Heawood noted that when doing a first recoloring the conditions under which the second recoloring can be carried out are destroyed. Simultaneous recolorings via Kempe s method are not possible; Heawood provided a counterexample to illustrate this point. More discussion of Kempe Chains will be included in Section Map-Coloring in Higher-Genus Spaces Map Coloring may also be considered on surfaces other than the plane (or, equivalently, the sphere). For any closed (orientable or non-orientable) surface with positive genus, the maximum number p of colors needed to color a map depends on the Euler Characteristic χ of the surface: p = χ 2 where the brackets represent the floor function. Note that the sphere is genus 0 and thus the above formula does not apply. There is one exception; the Klein Bottle (genus 2) has Euler Characteristic χ = 0 yet requires only 6 colors (vs. 7 according to the formula).

11 This is known as the Heawood Map-Coloring Conjecture and was proved by Ringel and Youngs in 1968 (Ringel and Youngs, 1968) The Five-Color Theorem In this section we prove the Five-Color Theorem. It is a far simpler proof than that of the 4CT yet the it employs many of the same ideas. Lemma If G is a connected simple planar graph with n ( 3) vertices and m edges, then m 3n 6 Proof. This lemma is a corollary to Euler s Formula (aka, Euler Characteristic) which states that for a connected plane graph G, n m+ f = 2 where n, m defined as above and f is the number of faces. Each face of G is bounded by at least 3 edges thus 3 f 2m (recall, each edge appears in 2 faces). Thus, combining this inequality with Euler s Formula we get n m + f = 2 6 = 3n 3m + 3 f 3n 3m + 2m m 3n 6. Lemma Every simple planar graph has a vertex of degree at most 5. Proof. Let G be a loopless planar graph. Without loss of generality, we may assume that G is connected and V(G) 3. Let n = V(G), m = E(G). Assume d G (v) 6 v V(G) then we have 6n 2m 3n m. Thus, it follows from Lemma that 3n 3n 6, a contradiction. Theorem (The Five-Color Theorem). Every loopless planar graph admits a vertex five-coloring. Proof. [Wilson] The proof is by induction on number of vertices. The result is trivial for graphs with fewer than six vertices. Suppose loopless planar graph G has n vertices and all loopless planar graphs with n 1 vertices are 5-colorable. Then G has a vertex v of degree at most 5 by Lemma We may delete v to obtain a graph G with

12 12 n 1 vertices. By the induction hypothesis we may find a five-coloring for G, call it c : V(G ) {1, 2, 3, 4, 5}. Now we wish to return v to the graph and find a coloring c : V(G) {1, 2, 3, 4, 5} such that c(u) = c (u) u V(G ) V(G). In other words, we wish to color v with one of the five colors. If d G (v) < 5 then we are done; let c(v) be any color k {1, 2, 3, 4, 5} such that c(u) k u adjacent to v. If d G (v) = 5 then let v 1,..., v 5 be the 5 adjacent vertices (clockwise ordering). Note, v 1,..., v 5 are not mutually adjacent for otherwise G would contain K 5, the complete graph on 5 vertices, as a subgraph but K 5 is not planar (K 5 is shown in Figure 2.1 below). Thus, at least two of these vertices not adjacent, say v 1 and v 2. Now, we contract edges vv 1 and vv 2, this new graph, G, has fewer than n vertices and is hence 5-colorable with c : V(G ) {1, 2, 3, 4, 5}. We reinstate these two edges and obtain the coloring c : V(G) {1, 2, 3, 4, 5} such that c(v 1 ) = c(v 2 ) = c (v). There are now at most 4 colors assigned to v 1,..., v 5, thus we may color v with a different color.

13 13 CHAPTER 2 The Four-Color Theorem There are several equivalent statements of the four-color theorem. We will be primarily concerned with the following: Theorem (Four-Color Theorem). Every loopless planar graph admits a vertex 4- coloring. The four-color theorem arose historically in a different form, related to the coloring of maps. We may formalize the definition of map as a 3-connected planar graph. A map G is k-colorable(f) if the faces (i.e., countries on the map) can be colored with k colors such that no two faces which share a boundary edge have the same color. When not clear from context, we will distinguish between face- and vertex-colorings by k- colorable(f) and k-colorable(v), respectively. The Four-Color Theorem is not true for non-planar graphs. Figure 2.1 below shows the complete graph on 5 vertices in which each vertex is adjacent to all four others; hence the graph is not planar and not 4-colorable(v). Figure 2.1: The complete graph on 5 vertices, K 5, is not planar and is not four-colorable.

14 CT for Maps and Graphs In this section we will show that the four-color theorem for maps, i.e. every map is 4- colorable(f), is equivalent to Theorem 2.0.4, the four-color theorem for graphs. First we must introduce the concept of the dual of a graph. This concept dates back at least to the time of Euclid Duals of Graphs Let G be a planar graph, then given a plane drawing of G we may construct its (geometric) dual G as follows: 1. Inside each face f of G choose a point v, these are the vertices of G. 2. For each edge e E(G) draw a line e that crosses only e and joins the vertices v in the 2 faces which border e, these are the edges of G. We now state some direct consequences of the definition of the dual. If G is plane and connected then so is G. Further, if G is plane and connected with n vertices, m edges, and f faces then G has f vertices, m edges, and n faces. If G is plane connected, then G is isomorphic to G. However, if G is isomorphic to H it is not necessarily true that G is isomorphic to H. An end vertex or bridge in G becomes a loop in G ; if two faces of G have more than one edge in common then G has multiple edges. Theorem Let G be a loopless plane graph. Let G be a geometric dual of G. Then G is k-colorable(vertex) if and only if G is k-colorable(face). Proof. Let G be simple and connected. Then G is a map. Let c : V(G) {1,..., k} be a k-coloring(v) for G. Let F(G ) be the faces of G, define c : F(G ) {1,..., k} as c ( f ) = c(v f ) where f F(G ) and v f is the unique vertex of V(G) in f. We claim this

15 15 defines a four-coloring(f) for F(G ). We only need to show that no two adjacent faces of G have the same color. But this follows from construction, for any two adjacent faces, say f 1, f 2 F(G ), c(v f1 ) c(v f1 ) c ( f 1 ) c ( f 1 ) Suppose we have a k-coloring(f) of G, c : F(G ) {1,..., k}. Then, by reasoning similar to above, we may define a k-coloring(v) on G, c : V(G) {1,..., k} by c(v f ) = c ( f ). Corollary The four-color theorem for maps is equivalent to the four-color theorem (for simple planar graphs). Proof. Let G be a simple connected plane graph. Then its dual G is a map which, by assumption, is four-colorable(f). Thus, by Theorem 2.1.1, G is four-colorable(v). Let G be a map and G be its dual. Then G is a simple planar graph and is, by assumption, four-colorable(v). Thus, by Theorem 2.1.1, G is four-colorable(f). Throughout this paper, for expository purposes certain ideas may be explained in terms of maps with countries (i.e., faces or regions) rather than graphs with vertices. 2.2 More Equivalent Conditions We have thus restated the traditional Four-Color Theorem in terms of vertex colorings of graphs. In the proof of Robertson et al we will encounter yet another equivalent statement of the 4CT (Theorem below). Before we proceed with examining their proof we state a few assumptions that we make without loss of generality. The 4CT is traditionally stated for graphs or maps on a plane. By a simple application of stereographic projection, we may alternatively consider the 4CT for graphs on the 2-sphere. We will use both conceptualizations herein. Below we will define triangulations and near-triangulations, the distinction between these two types of graphs will

16 16 illuminate the purpose for considering the 4CT on both the sphere and the plane. We will also show below that proving the 4CT for (near-)triangulations is sufficient for proving the 4CT in general. Likewise, we have limited ourselves to planar loopless graphs without multiple edges (i.e., simple planar graphs); when considering vertex colorings the condition to disallow multiple edges between vertices is implicit. 2.3 Overview of the Proof by Robertson et al The proof of Robertson et al employs the same basic strategy as that of Appel & Haken (A&H) though it is more streamlined and requires less checking by computer. They begin by defining a set of 633 configurations. None of these configurations may exist in a minimal counterexample to the 4CT; if one did appear then it could be replaced by something smaller, thus contradicting minimality. This section of the proof is called proving reducibility. In particular, every minimal counterexample is shown to be an internally 6-connected triangulation and it is proved that every internally 6-connected triangulation contains one of the 633 configurations. This is called proving unavoidability. Thus, no minimal counterexample exists and the four color theorem is true. Checking unavoidability uses the procedure of discharging vertices. It involves 32 discharging rules instead of 300+ by A&H and is checked by a computer. Robertson et al obtain a quadratic-time 4-coloring algorithm (vs. quartic-time by A&H). The focus of this paper will not be on the role of the computer or on the coloring algorithm but rather on the steps that lead up to and justify the use of a computer in the proof of the four-color theorem. The remainder of this paper is structured in accordance with the proof of Robertson et al. The next three sections examine Configurations, Reducibility, and Unavoidability respectively.

17 17 CHAPTER 3 Configurations We will consider the four-color problem on the 2-sphere and on the plane; as mentioned above these are equivalent conditions. When considering planar graphs there will be one region designated as the infinite region, all the others are finite. In figures (i.e., plane drawings) representing planar drawings the outside is considered infinite. 3.1 Triangulations A region (or face) of a plane graph is called a triangle if it is incident with exactly 3 edges. A connected, loopless plane graph on the sphere is called a triangulation if every region is a triangle. In particular, no circuit of length 2 bounds a region of a triangulation. A near-triangulation is a non-null connected planar graph G in which every finite region is a triangle. Figure 3.1: A triangulation. Note this graph is 4-colorable but not 3-colorable. Theorem To prove the 4CT for graphs on the 2-sphere, it is sufficient to prove the 4CT for triangulations. Proof. Let G be a graph on the 2-sphere S 2. Let F(G) be the faces of G. Then every f F(G) is a polygon projected onto the sphere. Say f is an n-gon, n > 3, with vertices v 1,..., v n, add vertex v f to the face of f and draw edges v i v f for i = 1,..., n. Do this for

18 18 each n-gon f F(G) such that n > 3 to obtain triangulation G. Suppose there exists a four-coloring(v) for c : V(G ) {1, 2, 3, 4}, then c restricts to a four-coloring on G. The following is an immediate consequence of Theorem 3.1.1: Corollary To prove the 4CT for graphs on the plane, it is sufficient to prove the 4CT for near-triangulations. 3.2 Minimal Counterexamples Suppose there is a map which is not four-colorable. Then there is such a map with the fewest number of countries. Such a map would be a minimal counterexample; any map with fewer countries can be colored with four colors. We wish to show that no minimal counterexamples exist. We define this concept in terms of vertex-colorings on planar graphs. A minimal counterexample is a plane graph G which is not 4-colorable such that every graph G with V(G ) + E(G ) < V(G) + E(G) is four-colorable. From this definition, we may show that every minimal counterexample is a triangulation. Theorem Every minimal counterexample is a triangulation. Proof. Let G be a minimal counterexample. If there is a region of G bounded by a circuit of length 2, say e 1 e 2 where e 1, e 2 E(G) are incident with vertices v and w then replace e 1 and e 2 with a single edge vw to obtain graph G. Clearly, this will not affect the vertex coloring of G but V(G ) + E(G ) < V(G) + E(G), a contradiction. Thus G cannot have a region bounded by a circuit of length 2. Now assume G has a region bounded by a circuit of length n 4. Let v i, i = 1,..., n be the n vertices on the boundary of the region. Without loss of generality, we may choose 2 of these vertices, say v 1, v 2 which are not adjacent. Draw edge v 1 v 2 and contract

19 19 it to a vertex v, call the new graph G. Graph G must be 4-colorable with coloring c : V(G ) {1, 2, 3, 4}. Thus, we may define a 4-coloring, c, on G by: c(u) = c (u) u V(G) V(G ) and c(v 1 ) = c(v 2 ) = c (v), a contradiction. Birkhoff proved additional properties of minimal counterexamples in 1913 (Birkhoff, 1913). He examined a particular type of circuit which he called a ring 1. Birkhoff s notion of a ring is a circuit with vertices {v 1,..., v n } such that v i is a neighbor of v j if and only if i j = 1 or n 1; the ring-size is n. Note that we will define ring and ring-size differently (in alignment with Robertson et al) in Section 4.2. By using a Kempe Chain argument he showed that a minimal counterexample can have neither: 1. a ring of size 4, nor 2. a ring of size 5 which contains more than one vertex. We prove Theorem below which implies (1) and(2). For a minimal counterexample G, there is an equivalent way to describe properties (1) and (2) from above. Robertson et al define a short circuit of a triangulation as a circuit C with E(C) 5, so that for both open discs on the sphere bounded by C, V(G), and V(G) 2 if E(C) = 5. In other words, a short circuit with 4 edges has vertices in both its interior and exterior. If it has exactly 5 edges then there are at least 2 vertices in the interior and exterior. We say G is internally six-connected if it has no short circuit. The above discussion gives us the following theorem. Theorem Every minimal counterexample is an internally 6-connected triangulation. Thus, there are relatively few types of circuits of short length in a minimal counterexample G. We have shown in Lemma that there are no circuits of length 2. 1 Actually, Birkhoff worked in terms of maps and regions. We consider the dual notions here.

20 There are plenty of circuits of length 3, namely triangles. Examples of permissible and non-permissible circuits (along with interior edges) of length 4 and 5 are shown in Figures 3.2 and 3.3. Every circuit of size 5 consists of either the vertices of triangle(s) or are the neighbors of a single 5-vertex. Figure 3.2: Permissible circuits of size 5 with interior edges. Figure 3.3: A short circuit with interior edges & vertex More on Kempe Chains A further examination of Figure 3.3 will be an instructive demonstration of the Kempe Chain method and will give further insight into the definition of an internally 6-connected triangulation. We first make precise the discussion of Kempe chains from Section A Kempe chain in a vertex 4-colored graph G is a sequence of vertices colored with only two colors. A Kempe chain in colors b, g {1, 2, 3, 4} is called a (b, g)-chain. A Kempe net is a component of the subgraph G bg, the subgraph of G spanned by all vertices colored b or g. Hence, each pair of vertices in a Kempe net can be joined by a Kempe chain. Our previous discussion of Kempe chains in Section describes the process of interchanging colors in a Kempe net. This process is called Kempe interchange and results in a new 4-coloring. Figure 3.3 has a 4-vertex (a vertex of degree 4). As it appears in a triangulation, then there must be a circuit of length four about it, i.e. a short circuit. Kempe proved that no such vertex can exist in a minimal counterexample using his method of Kempe Chains, we now present the proof. Theorem There is no vertex of degree 4 in a minimal counterexample.

21 21 Proof. Let y be a vertex of a minimal counterexample G such that d G (y) = 4. Let x 1,..., x 4 be the neighbors of y in cyclic order. Remove y and its incident edges from G to obtain graph G. Since G is a minimal counterexample there exists a vertex four-coloring for G, say c : G {1, 2, 3, 4}. If only three colors are needed for the neighbors of y then there is a 4 th color free for y and we may extend c to a 4-coloring of G. Thus we assume x 1,..., x 4 are colored with 4 colors. Without loss of generality, assume c (x i ) = i {1, 2, 3, 4}. We consider two cases: Case 1. If x 1 and x 3 are in different (1, 3)-nets then we may perform a Kempe interchange so that x 1 is now colored with color 3. Then y has neighbors of only colors 2, 3, 4 and c may be extended to a 4-coloring on G by defining c (y) = 1. Case 2. If x 1 and x 3 are in same (1, 3)-net then, by our discussion in Section 1.2.1, x 2 and x 4 must belong to different (2, 4)-nets and we may perform a Kempe interchange so that x 2 is colored with color 2. Thus we may again extend c to a 4-coloring of G. Kempe erroneously used a similar process to show that a minimal counterexample cannot have a five vertex. However, as discussed in Section 1.2.1, there was a flaw in his logic as he attempted to do two Kempe interchanges simultaneously. Nonetheless, Theorem proves that every vertex in a minimal counterexample has degree at least 5. Thus every (interior) vertex in a minimal counterexample is surrounded by a circuit of length at least 5. Thus Birkhoff showed that a minimal counterexample can have neither (1) a ring of size 4 nor (2) a ring of size 5 which contains more than one vertex as stated above. In the language of Robertson et al, it cannot have a short circuit. Robertson et al offer another approach to proving Theorem They are able to define a quadratic-time algorithm which constructs a 4-coloring of a triangulation G from a short circuit of G. The existence of such an algorithm and Theorem imply Theorem

22 Configurations A configuration K consists of a near-triangulation G(K) and a map γ K : V(G(K)) Z 0 with the following properties: 1. For every v V(G(K)), G(K)\{v} has at most two components, and if there are two then γ K (v) = d(v) For every v V(G(K)), if v is not incident with the infinite region, then γ K (v) = d(v). Otherwise, γ K (v) > d(v). In either case γ K (v) 5 3. K has ring-size 2 where the ring-size of K is v(γ K (v) d(v) 1), summed over all vertices v that are incident with the infinite region such that G(K) \ v is connected. Figure 3.4 shows examples of configurations. The vertex shapes represent the value of γ K at that vertex. The shapes follow the conventions established by Heesch in the 1960s and used by Robertson et al in their proof. For the examples below we have two vertex shapes, represents γ K (v) = 5 and. represents γ K (v) = 6... Figure 3.4: Examples of configurations. A subgraph G of G is induced if every edge of G with both ends in V(G ) belongs to G. Figure 3.5 shows an example of a subgraph which is not induced. A configuration K appears in a graph G if G(K) it is an induced subgraph of G, every finite region of G(K) is a region of G, and γ K = d G (v) v V(G(K)). This offers some more motivation for the definition of the configuration above, γ K (v) corresponds to the degree of v in the graph in which the configuration is induced. The first and second examples in Figure 3.4 differ in how they would appear in a graph.

23 23 G G Figure 3.5: Graph G is a subgraph of G but it is not an induced subgraph. We make some further observations about configurations. They are neartriangulations in which there are no bridges or vertices of degree 1. The outer vertices (those which are incident with the infinite region) form a closed circuit of size 4. Inner vertices always have degree 5. The ring-size condition (3) will allow us to prove the existence of free completions as defined in Section 4.2; this condition will be elaborated upon in the proof of Proposition Good Configurations Robertson et al define a set of 633 configurations. Two configurations K and L are isomorphic if there is a homeomorphism of the 2-sphere which maps G(K) to G(L) and γ K to γ L. A configuration isomorphic to one of these 633 configurations is called a good configuration. Figure 3.4 shows three good configurations. We now present two key theorems that comprise the proof of the four-color theorem. Theorem If T is a minimal counterexample, then no good configuration appears in T. Theorem For every internally 6-connected triangulation T, some good configuration appears in T. The proof of Theorem requires a computer. However, aspects of the proof will be examined in more detail in Section 4: Reducibility. Theorem can be checked by a computer in a matter of minutes; this theorem will be explored in Section 5: Unavoidability.

24 Tricoloring Let G be a triangulation or near-triangulation. A triangle r of G is tri-colored by a function κ : E(G) { 1, 0, 1} if {κ(e), κ( f ), κ(g)} = { 1, 0, 1} where e, f, g are the three edges incident with r. That is, edges which bound the same face are each colored differently. We say that κ is a tri-coloring of G if every (finite) region is tri-colored. The choice of { 1, 0, 1} to represent the colors is consistent with the work of Robertson et al and their computer programs. Note that the dual of a triangulation is a cubic graph, that is a regular graph of degree 3. Theorem below and Theorem imply that a triangulation is tricolorable if and only if it is four-colorable(v). First we prove a lemma. Lemma If the degree of each vertex in graph G has degree two, then G is twocolorable(f). Proof. Graph G is a collection of cycles. Each cycle encloses a single face and is bounded completely by the infinite region. Hence, we may color the infinite face with color 0 and color all other faces with color 1. Theorem Let G be a cubic graph. Then G is four-colorable(f) if and only if it admits a tri-coloring. Proof. [Wilson] Suppose G has a four-coloring(f) with coloring c : F(G) {α, β, γ, δ}. Denote the colors by α = (1, 0), β = (0, 1), γ = (1, 1), and δ = (0, 0). For e E(G), e belongs to two faces, say f 1, f 2, define κ : E(G) {α, β, γ} by κ(e) = c( f 1 ) + c( f 2 ), component-wise addition modulo 2. Note that δ cannot occur in this tri-coloring since the two end-vertices of any edge must be distinct. Thus, κ is a tri-coloring of G. Suppose κ : E(G) {α, β, γ} is a tri-coloring of G. Since G is a cubic graph, at each vertex there is an edge of each color. Let G α,β be the subgraph of G determined by

25 25 the edges colored α or β. Subgraph G α,β is regular of degree 2 and, by Lemma 3.5.1, G is 2-colorable(f) with colors {0, 1}. Let G α,γ be the subgraph of G determined by the edges colored α or γ. Likewise, G α,γ is regular of degree 2 and is thus 2-colorable(f) with colors {0, 1}. So at each face of G we may define two coordinates from these colorings. Since the coordinates assigned to adjacent faces must differ in at least one place the coordinates (1, 0), (0, 1), (1, 1), (0, 0) define a four-coloring. The preceding theorem (Theorem 3.5.2) is a result of Tait that dates back to Robertson et al found it easier to work with tri-colorings in their computer programs. Much of what follows will use this equivalent statement of the four-color theorem Contraction Theorem states that no good configuration appears in a minimal counterexample. The proof relies on the idea of contracting edges. If G is a minimal counterexample and we contract a non-null set of edges then we produce a new graph G. From minimality, G has a vertex four-coloring. In the proof, this coloring for G is converted into a fourcoloring for G. The process of contraction can get a little tricky. There are some potentially serious issues; for example, does G exist and is it loopless? There are also notational difficulties; for example, is an edge in G the same as the corresponding edge in G? Robertson et al pay particular attention to these difficulties in the statement and proof of Theorem below and in the defintions which precede it Tri-coloring of G Modulo X Let G be a triangulation or near-triangulation. A subset X E(G) is sparse if every region of G is incident with at most one edge in X and, in the case of a near-triangulation,

26 26 the infinite region is incident with no edge in X. If X E(G) is sparse, a tri-coloring of G modulo X is a map κ : E(G) \ X { 1, 0, 1} such that for every (finite) region incident with edges e, f, g the following conditions hold: 1. if e, f, g X, then {κ(e), κ( f ), κ(g)} = { 1, 0, 1} 2. if g X then κ(e) = κ( f ) Note, a tri-coloring is a tri-coloring modulo. The following theorem allows us to consider the graph G obtained by contracting all the edges of a sparse set X E(G), G a minimal counterexample, without having to explicitly define G. It allows us to avoid many notational difficulties. Theorem Let G be a minimal counterexample, let X E(G) be sparse, X, and suppose there is no circuit C of G such that E(C) X = 1. Then G admits a tri-coloring modulo X. Proof. Let F be a (possibly disconnected) graph with vertex set V(G) and edge set X. Let Z 1,..., Z k be the vertex sets of the k components of F. Now let S be the graph on k vertices such that V(S ) = {Z 1,..., Z k } and E(S ) = E(G) \ X. We say that e E(S ) is incident with Z i if e is incident with a vertex in Z i. Claim: S is loopless. Proof of Claim: Assume not. Then, without loss of generality, there is an e E(G) \ X with both end-vertices in Z 1. But then e plus some subset of X form a circuit C. But then E(C) X = 1, a contradiction. Thus, the claim is proved. So S is loopless and furthermore, since it is obtained from G by contracting edges in X, it is planar. Since X is non-empty we have V(S ) + E(S ) < V(G) + E(G) implies that S admits a vertex four-coloring (since G is a minimal counterexample). Thus we may extend this coloring to a map φ : V(G) {1, 2, 3, 4} such that 1. for 1 i k, φ(v) is constant for v Z i

27 27 2. for every edge e E(G) \ X with ends u and v, φ(u) φ(v) For each e E(G) \ X with ends u, v, define 1 if {φ(u), φ(v)} = {1, 2} or {3, 4}, κ(e) = 0 if {φ(u), φ(v)} = {1, 3} or {2, 4}, 1 if {φ(u), φ(v)} = {1, 4} or {2, 3}. We now check that κ : E(G) \ X { 1, 0, 1} is a tri-coloring of G modulo X. Let r be a (finite) region of G incident with edges e, f, g and vertices u, v, w, where e, f, g have ends uv, vw, uw respectively. If e, f, g X then φ(u), φ(v), φ(w) are all distinct. Thus, {κ(e), κ( f ), κ(g)} = { 1, 0, 1}. And if e X then φ(u) = φ(v) and so κ( f ) = κ(g). The above theorem (Theorem 3.5.3) will be applied to X which is the edge-set of a forest of T (i.e., no circuits). It is used in the proof of Theorem in Section 4.5 below. e 1 e 2 e 3 e 4 e 5 e 6 e 9 e 12 e 8 e 7 e 11 e 10 e 13 e 14 e 17 e 16 e 15 e 19 e 18 e 20 e 21 e 22 e 23 Figure 3.6: A near-triangulation G with no short circuit. Consider Figure 3.6 with the sparse set X = {e 9, e 11 }. According to Theorem a tri-coloring of G modulo X exists. By definition, for any such tri-coloring κ, it is required that κ(e 10 ) = κ(e 16 ) = κ(e 17 ), κ(e 2 ) = κ(e 5 ), and κ(e 3 ) = κ(e 6 ).

28 28 CHAPTER 4 Reducibility This section is largely devoted to developing additional machinery necessary for the proof of Theorem which states that good configurations do not appear in minimal counterexamples. We will conclude this section with a partial proof of this theorem, aspects of the proof which depend on a computer are omitted. 4.1 Consistent Sets of Edge Colorings Circuits will play an important role in working toward the proof of Theorem It is an obvious property of a near-triangulation that the infinite region is bounded by a closed walk which we will identify with a circuit. We will develop some important properties of this boundary circuit. An edge-coloring of a circuit R is a map κ : E(R) { 1, 0, 1}. It is clear, but worth noting, that a circuit is either 2- or 3-colorable, depending on whether it is of even or odd length. We wish to define a consistent set of edge colorings of a circuit R. First, some preliminary definitions are required. A match m in a circuit R is an ordered pair {e, f } of distinct edges of R. A signed match in a circuit R is a pair {m, µ} where m is a match and µ = ±1. A signed matching in a circuit R is a set M of signed matches, so that if ({e, f }, µ), ({e, f }, µ ) M are distinct then: 1. {e, f } {e, f } =, and

29 2. e, f belong to the same component of the graph obtained from R by deleting e and f. 29 If M is a signed matching then we denote by E(M) the set E(M) = {e E(R) : e m for some (m, µ) M} A set C of edge-colorings of a circuit R is consistent if κ C and θ { 1, 0, 1} a signed matching M such that κ θ-fits M, and C contains every edge-coloring that θ-fits M. a b e c d Figure 4.1: A circuit R. Consider circuit R in Figure 4.1. The set M = {{(a, b), 1}, {{(c, d), 1}} is a signed matching but N = {{(a, c), 1}, {{(b, d), 1}} is not because it violates condition (2) of the definition above. Let κ : E(R)) { 1, 0, 1} be defined by κ(a) = κ(c) = 1, κ(b) = κ(d) = 0, & κ(e) = 1. Then κ 1-fits M. By permuting the values of κ we would obtain a consistent set of edge-colorings. For any near-triangulation H there is a closed walk which traces the boundary of the infinite region (unique up to choice of initial vertex and orientation). Denote it by v 0, f 1, v 1,..., f k, v k = v 0 with f i E(H), v i V(H). We note that since H may not be 2-connected(v) this walk may have repeated vertices or edges. Let R be a circuit graph (not necessarily a circuit of H) of length k with edges e 1,..., e k in order. For 1 i k, define φ(e i ) = f i, then φ wraps R around H. If κ is

30 30 a tri-coloring of H then for e E(R) let λ(e) = κ(φ(e)) then λ is an edge-coloring of R called a lift of κ (by φ). Theorem Let H be a near-triangulation, and let φ wrap a circuit R around H. Let C be the set of all lifts by φ of tri-colorings of H. Then C is consistent. Proof. First note that the null set is consistent. Let e 1,..., e k, f 1,..., f k be defined as above in the definition of wraps. Thus, e i are the edges of a k-circuit R which wraps around H, a near-triangulation bounded by edges f i. Let λ C, λ : E(R) { 1, 0, 1}. Let θ { 1, 0, 1}. We must show C is consistent; i.e., that there is a signed matching M such that λθ-fits M and C contains every edge-coloring of R that θ-fits M. Without loss of generality, assume θ = 0. Since λ C, the set of all lifts by φ of tri-colorings of H, we know there is some tri-coloring κ of H for which λ is the lift. Let g 0, g 1,..., g t be distinct edges of H. Let r 1, r 2,..., r t be distinct finite regions of H. We define a rib to be a sequence such that g 0, r 1, g 1, r 2..., r t, g t 1. if t > 0 then g 0, g t are both incident with the infinite region of H, and if t = 0 then g 0 is incident with no finite region of H (recall the boundary of H may have repeated vertices or edges), 2. for 1 i t, r i is incident with g i 1 and with g i, and 3. for 0 i t, κ(g i ) 0. Thus, in any rib κ(g 1 ), κ(g 2 ),... is ±1 alternating and κ(e) = 0 e E(H) \ {g 0,..., g t } such that e is incident to some r i. Thus, if we reverse the signs of κ(g 1 ), κ(g 2 ),... we obtain a

31 31 new tri-coloring of H. As a consequence of the condition (2) in the above definition, we claim that any two ribs are disjoint (they share neither edges nor regions). Further, for 1 i k, f i belongs to a unique rib if κ( f i ) 0 and to no rib if κ( f i ) = 0. For each rib g 0, r 1, g 1, r 2..., r t, g t we associate the signed match ({e i, e j }, µ), where g 0 = f i and g t = f j and µ = +1 or 1 depending on whether or not t is even or odd, respectively (equivalently, whether κ(g 0 ) = κ(g t ) or not, respectively). To elaborate we recall that, for t > 0, g 0 and g t are both incident to the infinite region of H which is bounded by f l, 1 l k. We also recall that φ(e i ) = f i and φ wraps R around H. By construction, the set of all these signed matches is a signed matching M and λ θ-fits M. We still must show that C contains every edge-coloring of R that θ-fits M. Let λ be any such edge-coloring of R. We define κ ( f i ) = λ (e i ), 1 i k. This is welldefined: if f i = f j then λ(e i ) = λ(e j ) since λ(e) = κ(φ(e)) and thus, since λ also θ-fits M, λ (e i ) = λ (e j ). By reversing the signs of κ in some of the ribs we can construct a tri-coloring κ of H whose restriction to { f 1,..., f k } is κ. Thus λ is a the lift of κ and thus λ C. Proposition A set of edge-colorings S has a unique maximal consistent subset S. Proof. First note that the null set is consistent and that the union of any two consistent sets is consistent. Thus, the union of all consistent subset is nonempty and maximal. This maximal consistent subset can be found rather quickly using a computer for E(R) sufficiently small. The above theorems will be applied to good configurations which are small enough to make this computationally feasible. Computing this for E(R) = 14, the maximum which is needed for the proof of Robertson et al, takes less than a minute. Robertson et al have defined Algorithm

32 32 Algorithm An algorithmic version of Theorem Input: A near-triangulation T, a circuit R, a function φ which wraps R around T, and tri-coloring κ of T. Output: A set of tri-colorings of T including κ such that their lifts by φ are all distinct and form a consistent set. Running Time: O( V(T) ). 4.2 Free Completion Let K be a configuration. A near-triangulation S is a free-completion of K with ring R if 1. R is an induced circuit of S, and bounds the infinite region of S 2. G(K) is an induced subgraph of S, G(K) = S \ V(R), every finite region of G(K) is a finite region of S, and the infinite region of G(K) includes R and the infinite region of S, and 3. every vertex v of S not in V(R) has degree γ K (v) in S. Figure 4.2: A free completion of Figure 4.3 with ring-size 11. Figure 4.3: A Configuration K.Recall, vertex shape γ K (v) = 6, γ K (v) = 5 Recall that a subgraph G of G is induced if every edge of G with both ends in V(G ) belongs to G. The above definition gives insight into restrictions on γ K in the

33 33 definition of configuration given in Section 3.3; we are embedding the configuration K in a near-triangulation such that d S (v) = γ K (v). The ring insulates the configuration from the graph so that it may be reduced. Proposition Every configuration has a free completion. Proof. Let K be a configuration. Recall that the ring-size of K is v(γ K (v) d G(K) (v) 1), summed over all vertices v that are incident with the infinite region such that G(K) \ v is connected. If K appears in graph T then γ K (v) = d T (v) v V(G(K)). Thus, the ring-size of K is the length of ring R in its free completion S. And the condition that K has ring-size 2 implies that E(R) is non-null. We restate an important insight from the above proof as a lemma: Lemma Let K be a configuration. The ring-size of K is the length of ring R in its free completion S. If S 1, S 2 are both free-completions of K, then there is a homeomorphism which fixes G(K) and maps S 1 to S 2, free completions are unique up to homeomorphism and we may talk of the free completion. This is a consequence of condition (1) in the definition of configurations which states that v V(G(K)), G(K) \ v has at most two components. For if not, then there would be more than one choice for d S (v) in the completion D-reducibility Let S be the free-completion of a configuration K with ring R. Let C be the set of all edge-colorings of R, and let C C be the set of restrictions to E(R) of tri-colorings of S. Let C be the maximal consistent subset of C \C. Then we say that the configuration K is D-reducible if C = ; i.e., the colorings of the bounding ring correspond to the colorings of the configuration in a precise way. We will later see that no D-reducible configuration appears in a minimal counterexample (Theorem below). The sets

34 34 C and C are quite large; checking that a configuration is D-reducible is a task that Robertson et al accomplish with the use of a computer (more discussion of this follows). D-reducible configurations correspond to those which can be dealt with by the original methods of Kempe Contracts We use the same notation as above and define a special case of what is more generally known as C-reducibility. Let X E(S ) \ E(R). We say X is a contract for K if X, X is sparse in S, and no edge-coloring in C is the restriction to E(R) of a tri-coloring of S modulo X. Recall, a subset X E(G) is sparse if every region of G is incident with at most one edge in X and the infinite region is not incident with any edge in X. We will later see that no configuration K with a contract can appear in a minimal counterexample as long as the following condition holds: if K occurs in an internally 6-connected triangulation T then there is no circuit C of T with E(C) X = 1 where X is the set of edges of T corresponding to X. The general idea is to replace or contract parts of the configuration inside the ring R in order to obtain a coloring which may be extended to the configuration. In the appendix of their proof, Robertson et al identify 633 configurations. Certain of these are drawn with extra half-edges that indicate edges of the free completion of K. Some edges and half-edges are drawn thicker in the appendix. Robertson et al checked the following theorem by computer (it took about 3 hours on a Sun Sparc 20 workstation with 1 MB of RAM). It asserts that each of the 633 configurations is either D reducible or there exists a contract for K. Theorem For each of the 633 configurations K identified by Robertson et al, let X be the set of edges of the free completion of K drawn thickened in the figure. If X =, then K is D reducible. Otherwise, 1 X 4, and X is a contract for K.