Coloring Squared Rectangles

Transcription

1 Coloring Squared Rectangles Mark Bun August 28, 2010 Abstract We investigate the 3-colorability of rectangles dissected into squares. Our primary result is a polynomial-time algorithm for deciding whether a member of a large class of squared rectangles is 3-colorable, which runs in linear time for most squarings. We also give partial results in using the graph of the electrical network corresponding to a squared rectangle to decide its 3-colorability. Contents 1 Introduction 2 2 Preliminary Definitions 2 3 Graphs with Short Cycles Partial Wheels and Wheels Near-Triangulations A Crude Coloring Algorithm Coloring the Adjacency Graph 7 5 Colorability from the P-net Coloring Medial Graphs

2 1 Introduction Graph coloring problems are of great interest to mathematicians and computer scientists. One of the most significant achievements in twentieth century mathematics was the 1976 proof of the four-color theorem any planar graph admits a vertex coloring with four colors. Despite this positive result, it remains NPcomplete to decide whether four is the fewest number of colors needed to color a planar graph, or if three suffice. Of course, there are certain proper subclasses of the planar graphs for which the problem of 3-colorability is still NP-complete, as well as subclasses for which the problem is very easy. Examples of each are given in this paper. Another problem, reminisicent of recreational mathematics, is that of squaring the rectangle. Given a rectangle, the problem is to dissect it into a finite number of non-overlapping squares called elements. A dissection into n squares is called a squaring of order n. This problem becomes interesting when we place certain restrictions on the nature of the squaring a perfect squaring is one where each element has a different size, and a compound squaring is one where some proper subset of elements is itself a rectangle. A squaring that is not compound is simple. The main theoretical insight on squared rectangles, presented in [2], came from four undergraduates at Cambridge. The Cambridge students associated each squared rectangle to a discrete electrical network. Subsequent papers have shown that this seems to be the right way to think about the problem. In this paper, we merge ideas from these two problems and examine the colorability of squared rectangles. We first give an algorithm that decides whether a squared rectangle admits a 3-coloring. We then investigate whether we can use the underlying electrical network to solve this problem combinatorially. 2 Preliminary Definitions A plane graph G = (V, E) is a graph that has been embedded in the plane. That is, its vertices V are points in the plane and its edge set E consists of non-interesecting simple curves. A planar graph is one which is isomorphic to a plane graph. A simple graph has no self-loops or multi-edges (more than one edge between a pair of vertices). A k-coloring of a graph G = (V, E) is a mapping f : V {1,..., k} such that if G has an edge from u to v, then f(u) f(v). Sometimes we will take the codomain of f to be an actual set of colors. We say that G is k-colorable if it admits a k-coloring. A k-coloring f : V {1,..., k} is unique if every k coloring of G takes the form σ f for some permutation σ of {1,..., k}. The chromatic number χ(g) is the smallest k such that G is k-colorable. By a region of R 2, we mean an arc-connected subset. A face of a plane graph G is a maximal region R such that any pair of points on the interior of R is connected by a path that does not intersect an edge of G. A finite plane graph has exactly one unbounded face, which we will call its outer face. 2

3 Let v be a vertex of a graph G. The closed neighborhood N G [v] is the induced subgraph of G on v and its neighbors. Henceforth, we will simply use the term neighborhood and drop the subscript G when the graph in question is known. Finally, let d G (v) denote the degree of a vertex v in G. 3 Graphs with Short Cycles 3.1 Partial Wheels and Wheels In the next section, we will look at near-triangulations, which form a broad class of graphs which are easy to color. The building blocks of these graphs are partial wheels and wheels, in the sense that the neighborhood of each vertex in a near-triangulation is one of these objects. We present a few results on partial wheels and wheels which will simplify the proofs in later sections. Interestingly, wheel graphs create a subtlety in the algorithmic enumeration of simple squared rectangles; see [6] which points to the original theorem in [7]. Definition 1. Let n 3. A partial wheel of order n is a planar graph consisting of a hub vertex v connected to vertices v 1,..., v n, with additional edges v i v i+1 for 1 i < n. A wheel of order n is a partial wheel of order n where the vertices v 1 and v n are identified. Figure 1: Partial wheel and wheel of order 7 Lemma 3.1. Let P n be a partial wheel of order n. Then P n has a unique 3-coloring f where f(v) = 1 and { 2 if i is odd f(v i ) = 3 if i is even. 3

4 Proof. We uniquely determine the 3-coloring f. Suppose without loss of generality that f(v) = 1, f(v 1 ) = 2. The vertices of vv 1 v 2 must have distinct colors, so we must set f(v 2 ) = 3. Similarly, the vertices of vv 2 v 3 must have distinct colors, so we are forced to set f(v 3 ) = 2. The claim follows by induction on i. Corollary 3.2. Let W n be a wheel of order n. If n is even, then W n is not 3-colorable. If n is odd, then W n has a unique 3-coloring. Proof. Let n be even and suppose W n has a 3-coloring f. Then by Lemma 3.1, f(v 1 ) = 2 while f(v n ) = 3, which is a contradiction since we identify v 1 with v n. If n is odd, we can safely do this identification and the result in Lemma 3.1 carries over exactly. 3.2 Near-Triangulations Versions of the definitions and results in this section are given in [5]. Definition 2. A triangulation is a plane graph where every face is a triangle (bounded by a 3-cycle and having no vertices in its interior). Definition 3. A near-triangulation is a biconnected simple plane graph where all but at most one face is a triangle. Without loss of generality, we can take the nontriangular face of a neartriangulation G to be the outer face. The biconnectivity of G implies that every face, in particular the outer face, is bounded by a simple cycle. Call the vertices on the outer cycle the boundary vertices of G, and let the remaining vertices be interior vertices. We make one more definition: Definition 4. A near-triangulation is internally even if every interior vertex has even degree. We are now ready to give our most useful result on coloring. Theorem 3.3. Let G be a near-triangulation. i. If G is not internally even, then χ(g) = 4. ii. If G is internally even, then G has a unique 3-coloring. This is an extension of a result in [5], which follows from Heawood s corresponding result for triangulations. Unfortunately, I was unable to locate Heawood s paper, so this proof will have to suffice. Proof (i). Suppose G is not internally even. Let v be an interior vertex of G with degree 2n + 1. Since G s interior is triangulated, the neighborhood N[v] is a wheel of order 2n + 2. By Corollary 3.2, N[v] is not 3-colorable, so G is not 3-colorable. However, by the four-color theorem, G is 4-colorable so χ(g) = 4. 4

5 The proof of the second part is by way of a coloring algorithm that builds up 3-colored subgraphs of G. To simplify the algorithm, we define the primary operation we will use to construct these subgraphs. Definition 5. Let G be a near-triangulation with m-valent boundary vertex v. Suppose v is adjacent to boundary vertices u and w so that u, v and w are in clockwise order when traversing the boundary cycle of G. A k-interiorization G of G with respect to v is a graph obtained by adding vertices v 1,..., v k m in clockwise order on some arc from u to w in the outer face of G, as well as the edges wv 1, v k m u, v i v i+1 for 1 i < k m and vv j for 1 j k m. Observe that v is no longer a boundary vertex of G, but each v i, u and w are. Lemma 3.4. Let G be a near-triangulation with unique 3-coloring f. Let G be the graph obtained by 2k-interiorizing some boundary vertex v of G. Then G is a near-triangulation with a unique 3-coloring. Proof. The resulting graph G is clearly biconnected, and since the added faces are triangular, G is a near-triangulation. The neighborhood N G [v] is a wheel graph of order 2k + 1, so by Corollary 3.2, it has a unique 3-coloring. Since the colors of N G [v] must match up with the colors of v and its neighbors in G (of which there are at least 2), this extended coloring is completely determined by f(g). Observe that Lemma 3.4 holds in slightly more generality when we consider partial interiorizations. That is, instead of replacing the neighborhood of a vertex v with a wheel graph, we replace it with a partial wheel graph so that it remains a boundary vertex. A slight ammendment of the proof of Lemma 3.4 gives the same result for all (i.e., not necessarily even) partial interiorizations of a boundary vertex. Lemma 3.5. Let G and H be near-triangulations where H is a strict subgraph of G such that G \ H has no vertices or edges in the interior of H. Then there is a boundary vertex of H that can be interiorized or partially interiorized to give a strictly larger subgraph of G. Proof. Let v 1 be a boundary vertex of H where d H (v 1 ) < d G (v 1 ). Such a vertex exists since H is a strict subgraph of G and the interior vertices of H have exactly the same degree as they do in G. Suppose d G (v 1 ) = k. If the k-interiorization of H (or some partial interiorization) with respect to v 1 is a subgraph of G, we are done. Otherwise, there must be some boundary vertex of H, say v m, where v 1 v m is an edge in G but not an edge in H. This edge partitions off some cycle C = v m, v 1, v 2,..., v m 1, v m, v 1 on the interior of G, where v 1, v 2,..., v m are consecutive boundary nodes of H. Since the interior of G is triangulated, G has some edge v i v i+2. If we produce a new graph H by adding v i v i+2 to H, then v i+1 is an interior vertex of H. Hence H is a (nearly trivial) interiorization of H with respect to v i+1. Proof (ii). By induction on Lemma 3.5, we can form G by starting with K 3 and repeatedly applying interiorizations and partial interiorizations. If we color the 5

6 graph in each iteration according to Lemma 3.4, the final result is the unique 3-coloring of G. 3.3 A Crude Coloring Algorithm Consider a simple cycle C n of length n. We define a set of graphs T χ (n), the so-called χ-triangulations of C n, recursively as follows. 1. T χ (3) = {C 3 }. 2. The wheel W 5 is in T χ (4). 3. Let v 1 and v 2 be two non-adjacent vertices in C n ; connect them with an edge. This induces two shorter cycles C i and C j. Apply a triangulation from T χ (i) to C i and a triangulation from T χ (j) to C j. The resulting graph is in T χ (n). We now justify the name of this family of sets. First, it is immediately clear that each graph in T χ (n) is a near-triangulation. The next lemma gives the connection with coloring. Lemma 3.6. Let f be a 3-coloring of C n for n 3 (obviously at least one exists). Then there is member of T χ (n) with a 3-coloring that agrees with f on C n. Proof. Label the vertices of C n, in order, as v 1,..., v n. If n = 3, we are done. Suppose n = 4. Then one 3-coloring of C 4 is f(v 1 ) = f(v 3 ) = 1 and f(v 2 ) = f(v 4 ) = 2. In this case, W 5 T χ (4) and we can color the hub v in W 5 with color 3. The other 3-coloring is given by f(v 1 ) = 1, f(v 2 ) = 2, f(v 3 ) = 3 and f(v 4 ) = 2. The graph obtained by connecting v 1 to v 3 is in T χ (4), so we are done with this case. Now suppose n 5. Suppose as our inductive hypothesis that Lemma 3.6 holds for 3 k < n. We claim that there is a non-adjacent pair of vertices on C n that have different colors. Suppose f(v 1 ) = 1 and f(v 2 ) = 2. Then f(v 3 ) = 1 or 3. If f(v 3 ) = 3, then v 1 and v 3 have different colors as desired. If f(v 3 ) = 1, then f(v 4 ) = 2 or 3. In either case, v 1 and v 4 have different colors. Note that since n 5, the vertices we have produced are non-adjacent. Connecting these vertices produces two smaller cycles. By the inductive hypothesis, each has a χ-triangulation with a 3-coloring that agrees with f. By our definition, the graph resulting from these triangulations is in T χ (n). The consequence is that in a 3-colorable graph G where the boundary of every face is a cycle, there is a 3-colorable near-triangulation obtained by χ- triangulating each face of G. More formally, we give the following algorithm for deciding whether, say a biconnected graph, is 3-colorable: 6

7 Algorithm 1 3-colorability of biconnected planar graph G number the interior simple cycles of G of length 4 with i {1,..., n} let k i be the length of cycle i for H T χ (k 1 ) T χ (k 2 ) T χ (k n ) do set G to be G with its cycles replaced by the components of H if G is internally even then return accept end if end for return reject This algorithm takes an exponential (2 O(nc) for constant c) amount of time for general planar graphs: For each additional simple cycle of length k in G, the number of times the algorithm checks internal evenness increases by a factor of T χ (k). The number of χ-triangulations of C k is (tightly) bounded from below by the number of triangulations of a convex polygon on k vertices, which is known to be the (k 2)nd Catalan number. The Catalan numbers increase exponentially, which gives this running time. However, if we restrict the graphs we examine to those with few cycles of constantly bounded length, this algorthm becomes reasonably efficient. Lemma 3.7. Algorithm 1 runs in polynomial time for planar graphs on n vertices where 1. The length of each interior simple cycle is bounded by a constant k. 2. The number of interior simple cycles of length > 3 is O(log n). Proof. Let b = T χ (k). Then the number of graphs H we need to check in Algorithm 1 is bounded by for some constant c. b O(log n) = n c log b 4 Coloring the Adjacency Graph With these results on near-triangulations in hand, it is easy to analyze the behavior of squared rectangles by examining the properties of equivalent graphs. Definition 6. Let R be a squared rectangle. The adjacency graph A(R) has a vertex for every element in R, and an edge between two vertices if the corresponding elements in R are adjacent. Clearly R is k-colorable if and only if A(R) is also k-colorable. Obviously A(R) is a connected planar graph, so by the four-color theorem, a squared rectangle is always 4-colorable. 7

8 Definition 7. Let R be a squared rectangle. A cross of R is an intersection point of four of its elements. A cut element of R is an element whose removal disconnects R. Theorem 4.1. Let R be a squared rectangle with no crosses or cut elements. Then A(R) is a near-triangulation. Proof. The exclusion of cut elements implies that the removal of any single vertex from A(R) results in a a connected graph, so A(R) is biconnected. The boundary elements of R form a cycle, which in turn induces the outer face of A(R). We now show that the outer face is the only non-triangular face of A(R). Suppose X and Y are adjacent elements of R, and without loss of generality, X is stacked on top of Y. Further suppose that at least one has a left side that does not coincide with the boundary of R, and at least one has a right side that does not coincide with the boundary of R. Consider the interface between the left side of X and the left side of Y. Up to relabeling, we have two cases: Case 1: The left side of X and the left side of Y form a single vertical line segment. Suppose the left side of X does not coincide with the boundary of R, so there is at least one element of R adjacent to the left side of X. Let Z be the lowest such element. Suppose Z is not adjacent to Y. Then the bottom side of Z forms a single horizontal line segment with the bottom side of X, which clearly induces a cross. Hence Z is adjacent to Y, so there is a triangular face XY Z in A(R). Figure 2: Case 2 Case 2: The left side of X is strictly to the left (or to the right) of the left side of Y. In this case, some element Z must occupy the corner indicated in Figure 2. In this case as well, X, Y and Z form a triangle in A(R). We can repeat this analysis for the right sides of X and Y. The result is that the edge XY in A(R) is incident to two triangular faces. Via rotation, this shows that every edge of A(R) not on the boundary cycle is also incident to two triangular faces. Now consider the case where X and Y are adjacent on the boundary of R; say their right sides both coincide with the right side of R. Since we are disallowing cut elements, either Case 1 or Case 2 applies to the left sides of X and Y. Therefore, the interior face incident to XY on A(R) is a triangle, and the other face is just the outer face. 8

9 Corollary 4.2. If R is a squared rectangle with no crosses or cut elements, then A(R) is colorable if and only if it is internally even. It turns out that dealing with cut elements does not make our coloring problem significantly harder. A cut element of a squared rectangle must either coincide with both the top and bottom boundaries, or with both the left and right boundaries. In any particular squaring, it is impossible to have cut elements of both types. Therefore, without loss of generality, we can assume that the cut elements of a squared rectangle all coincide with its top and bottom. This translates to an easy dissection of the corresponding adjacency graph into biconnected components as in Figure 3. Figure 3: A(R) split into biconnected components As a consequence of our ability to express A(R) as this chain of biconnected components, A(R) is 3-colorable if and only if each component is 3-colorable. Since locating biconnected components takes polynomial time, a polynomialtime algorithm for biconnected graphs translates into one for general graphs. Now we handle the case where R has no cut vertices, but perhaps has crosses. By the proof of 4.1, the crosses in R are in direct correspondence with interior 4-cycles in A(R). Hence one option for deciding the 3-colorability of A(R) is to use our techniques from section 3.3. However, we can also define an even broader class of squared rectangles for which we can determine 3-colorability in polynomial time. Theorem 4.3. Consider the class of squared rectangles where each element is incident to at most two crosses. We can decide 3-colorability for this class in polynomial time. Proof. The proof is again by way of an algorithm. We build a boolean formula ψ in conjunctive normal form which is satisfiable if and only if some χ-triangulation of G is internally even. Fix a numbering C 1,..., C k of the 4-cycles of G = A(R). For each C i, associate variables x i and y i which represent the cycle s diagonals. This induces a correspondence between the χ-triangulations of C i and boolean formulas as in Figure 4. Hence C i has been χ-triangulated if and only if x i y i is satisfied, so we start by letting φ(x 1, y 1,..., x k, y k ) = i (x i y i ). 9

10 (a) x ȳ (b) x y (c) x y Figure 4: Correspondence with formulas Let G (x 1, y 1,..., x k, y k ) be the χ-triangulation of G corresponding to a satisfying assignment to φ. Internal evenness imposes additional logical constraints in terms of these variables. Set ψ = φ. Consider an interior vertex v. We have three cases. Case 1: Vertex v is incident to no 4-cycles. Then if our χ-triangulation is internally even, d G (v) must be even. We can therefore append 1 to (AND 1 with) ψ if d G (v) is even and 0 to ψ if it is odd. Case 2: Vertex v is incident to exactly one 4-cycle. Suppose the diagonal of this 4-cycle that is incident to v is z i. If d G (v) is even, then d G (v) is even if and only if z i = 0, so we append z i to ψ. Similarly, if d G (v) is odd, we append z i to ψ. Case 3: Vertex v is incident to exactly two 4-cycles. Let z i and w i be the diagonals incident to v. If d G (v) is even, then d G (v) is even if and only if z i = w i, or equivalently, (z i w i ) ( z i w i ). If d G (v) is odd, we append z i w i, or (z i w i ) ( z i w i ). The result is a 2-CNF formula ψ which is satisfiable if and only if G is internally even. Since ψ has a polynomial number of clauses, and 2SAT P, we can determine whether G is 3-colorable in polynomial time. Since simple, perfect squarings tend to have very few crosses, this result is probably sufficient for all intents and purposes with these squarings. The restriction on the number of crosses incident to each element seems necessary for this proof method to work. If an element is incident to 3 or 4 crosses (clearly 4 is the maximum), then the corresponding CNF we must append to ψ has no obvious equisatisfiable 2-CNF counterpart. It remains possible that coloring general squared rectangles is NP-complete; it would be of great interest to determine whether this is the case. 5 Colorability from the P-net In [2], the Cambridge students observed a correspondence between squared rectangles and resistor networks called p-nets (short for polar nets). The underly- 10

11 ing graph of a p-net consists of two boundary vertices called poles connected through interior vertices. The edges of the graph correspond to unit resistors. A positive voltage is impressed at one pole and a voltage of zero is impressed at the other. The resulting currents through the resistors are the sizes of the elements in the corresponding squared rectangle. Let P be a p-net and let R(P ) be its corresponding squared rectangle. Since P provides a succint representation of R(P ), we would ideally like to directly use P to determine the 3-colorability of R(P ). That is, while we could just compute currents to obtain R(P ) from P and use the results of the previous section, it would be interesting to see what information we can get just combinatorially. Our primary tool for understanding P is its medial graph. We follow the construction in [3]. Let M(P ) be the simplified medial graph of P, which is obtained from the medial graph of P by contracting its boundary spikes and removing its parallel edges and self-loops. Theorem 5.1. Let P be a p-net in the normal form of [2] corresponding to squared rectangle R. Then M(P ) is a subgraph of A(R) with an identical vertex set. Proof. Since P is in normal form, its currents are all nonzero so R has an element, and hence A(R) has a vertex, for every edge in P. Let uv be an edge in M(P ). Then the edges u and v are adjacent in a simple cycle of P. The simple cycles of P compress to vertical line segments in R, so u and v are adjacent in R and therefore A(R). Implicit in this result is that, in the case where R has no crosses, A(R) is just a near-triangulation of M(P ). The extra edges come from squares that are adjacent via the vertical line segments in R. As far as I know, the only way to determine these additional adjacencies is to compute the currents in P. Since the process of simplifying a medial graph does not affect 3-colorability, we have Corollary 5.2. If the medial graph of P is not 3-colorable, then R(P ) is not 3-colorable. 5.1 Coloring Medial Graphs We would like to be able to use Corollary 5.2 to give an easily computable necessary condition for 3-colorability of a squared rectangle. Unfortunately, there probably isn t one. The theorem in this section is a consequence of the following result from [4]. Lemma colorability of 4-regular simple, planar graphs is NP-complete. The proof of this fact itself follows from a chain of reductions given in [4] and its references. Theorem 5.4. For k 0, 3-colorability is NP-complete for medial graphs on exactly 2k 1-valent nodes. 11

12 Proof. Like almost all 3-coloring problems, this one is obviously is in NP. Fix k 0. We give a polynomial-time reduction that makes use of Lemma 5.3. Let G be a 4-regular simple, planar graph. In linear time, we can compute a planar embedding for G (consult, for instance, [1]). Let v be a vertex on the outer face of G. Without loss of generality, v and its four (distinct) neighbors are positioned as in Figure 5. Figure 5: Configuration before transformation Figure 6: Configuration after transformation Let G be the graph obtained by replacing the configuration in Figure 5 with the one in Figure 6. If G is 3-colorable, then if we let the green in Figure 6 represent the color assigned to v, we have a 3-coloring of G as well. It is easy to see that the red, yellow, and green 3-coloring of the gadget in Figure 6 is unique; in particular, the green vertices must all have the same color. Therefore, a 3- coloring of G can be compressed into a 3-coloring of G. Note that G has two additional 4-valent nodes on its exterior face. Hence, we can repeatedly apply this transformation to these new exterior vertices k times. This yields a medial graph on 2k 1-valent vertices that is 3-colorable if and only if G is 3-colorable. 12

13 References [1] J. M. Boyer and W. J. Myrvold, On the cutting edge: simplified O(n) planarity by edge addition, J. Graph Algorithms Appl. (3) 8 (2004) [2] R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J. 7 (1940) [3] E. Curtis and J. Morrow, Inverse Problems for Electrical Networks, Series on Applied Mathematics, World Scientific, [4] D. P. Dailey, Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete, Discrete Math. 30 (1980) [5] K. Diks, L. Kowalik, and M. Kurowsi, A new 3-color criterion for planar graphs. (English summary) Graph-theoretic concepts in computer science , Lecture Notes in Comput. Sci., 2573, Springer, Berlin, [6] N. D. Kazarinoff and R. Weitzenkamp, Squaring Rectangles and Squares. Amer. Math. Monthly (8) 80 (1973) [7] W. T. Tutte, A theory of 3-connected graphs. Indag. Math. 23 (1961)