Graphs and Combinatorics (2007) 23:87–95 Digital Object Identifier (DOI) 10.1007/s00373-006-0687-7
Graphs and Combinatorics © Springer-Verlag 2007
3-Colorable Even Embeddings on Closed Surfaces Atsuhiro Nakamoto1 and Norihito Sasanuma1 1 Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan. e-mail:
[email protected] Abstract. It is known that for any closed surface F 2 , every even embedding on F 2 with sufficiently large representativity is 4-colorable. In this paper, we shall characterize 3-colorable even embeddings on F 2 with sufficiently large representativity. Key words. Even embedding, Cycle parity, Coloring, Representativity
1. Introduction We consider only a fixed embedding of some simple graph on a closed surface F 2 , which is called an embedding on F 2 . A cycle C is said to be even (resp., odd) if the length of C is even (resp.,odd). An embedding on F 2 is said to be even if each face is bounded by an even cycle. A quadrangulation on F 2 is an even embedding on F 2 such that each face is quadrilateral. A triangulation on F 2 is an embedding on F 2 such that each face is triangular. We denote the orientable closed surface of genus g and the nonorientable closed surface of crosscap number k by Sg and Nk , respectively. A closed curve on a closed surface F 2 is said to be essential if does not bound a 2-cell. We say that a closed curve is 1-sided if the tubular neighborhood of is ¨ homeomorphic to the Mobius band, and is 2-sided otherwise. The representativity of a graph G on F 2 , denoted by r(G), is defined as the minimum number of intersecting points of G and a closed curve on F 2 , where ranges over all essential closed curves. In this paper, we often cut a closed surface F 2 in which a graph G embeds along a cycle C of G or a simple closed curve on F 2 intersecting only vertices of G. The cutting along C or means a cutting of F 2 into a punctured surface. Note that after cutting, all vertices on C or appear twice on its boundary component(s). Let G be a graph. A map c : V (G) → {1, . . . , k} is called a k-coloring of G if for any edge xy of G, c(x) = c(y). We say that G is k-colorable if G admits a k-coloring. The chromatic number of G, denoted by χ (G), is the smallest integer k such that G is k-colorable. We say that G is k-chromatic if χ (G) = k.
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Theorem 1 (Hutchinson [2]). Let G be an even embeddings on closed surfaces F 2 . Then 5 + 25 − 16ε(F 2 ) χ (G) ≤ . 2 Moreover, these bounds are sharp unless F 2 is the Klein bottle N2 or the double torus S2 . However, even embeddings with sufficiently large representativity do not need so many colors. In particular, Hutchinson has proved the following theorem. Theorem 2 (Hutchinson [3]). For any orientable closed surface Sg with g > 0, there exists a positive integer M1 (S) such that every even embedding G on Sg with r(G) ≥ M1 (S) is 3-colorable. However, such a theorem does not hold for any nonorientable closed surfaces. The following theorem is known for the projective plane. Theorem 3 (Youngs [8]). Every non-bipartite quadrangulation on the projective plane is 4-chromatic. Moreover, Mohar and Seymour [4] and Nakamoto, Negami and Ota [6] have independently given characterizations of 4-chromatic even embeddings on nonorientable closed surfaces with sufficiently large representativity, as follows. Theorem 4 (Mohar and Seymour [4], and Nakamoto, Negami and Ota [6]). For any natural number k, there exists a positive integer M2 (k) such that every even embedding on Nk with representativity at least M2 (k) is 4-colorable. In particular, G is 4-chromatic if and only if (i) G has an odd cycle cutting Nk into an orientable surface, and (ii) G has a quadrangulation as a subgraph. In this paper, we shall give a characterization for 3-colorable even embeddings on closed surfaces, which is a complementary result of Theorem 4. Let G be an embedding on a closed surface F 2 and let be a simple closed curve on F 2 where is G embedded. We say that is a bipartizing if the graph obtained from G by removing all vertices and edges intersecting is bipartite. We also say that is independent in G if intersects G only at their vertices and the vertices of G through which γ passes are independent in G. The following is our main theorem. Theorem 5. For any closed surface F 2 , there exists a positive integer N (F 2 ) satisfying the following: Let G be an even embedding on F 2 with representativity at least N(F 2 ). Then G is 3-colorable if and only if G has a bipartizing independent curve. In particular, no restriction for the representativity is needed in the projective planar case.
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Fig. 1. A set of generators for Sg
2. Cycle Parities In this section, in order to prove our main theorem, we shall introduce a homological invariant for even embeddings, called cycle parity, which was first defined in [5]. Let π1 (F 2 ) be the fundamental group of a closed surface F 2 . We call any homomorphism ρ : π1 (F 2 ) → Z 2 a cycle parity over F 2 . A closed curve is said to be even (rep., odd) under a cycle parity ρ if ρ([]) = 0 (resp., = 1), where [] denotes the homotopy class corresponding to . Two cycle parities ρ and ρ are said to be congruent if there is a homeomorphism h : F 2 → F 2 which induces an automorphism h∗ : π1 (F 2 ) → π1 (F 2 ) with ρh∗ = ρ . To express cycle parities as vectors, we fix a system {a1 , b1 , a2 , b2 , . . . , ag , bg } of simple closed curves on the orientable closed surface Sg so that for i = 1, . . . , g, ai and bi cross exactly once transversely and each of ai and bi crosses no other aj and bj with i = j . (For example, see Fig. 1.) It is easy to see that any cycle parity ρ can be determined by ρ(a1 ), ρ(b1 ), . . . , ρ(ag ), ρ(bg ). Thus, we denote ρ as the sequence of these value, that is, we write ρ = (ρ(a1 ), ρ(b1 ), . . . , ρ(ag ), ρ(bg )), which is a sequence of 0’s and 1’s. In particular, ρ = (0, 0, . . . , 0) is said to be trivial. Theorem 6 ([6]). Any non-trivial cycle parity ρ on the orientable closed surface Sg is congruent to (1, 0, . . . , 0). In order to express cycle parities over N2k+1 , we take a system {x, a1 , b1 , . . . , ak , bk } of simple closed curves on N2k+1 such that x is a 1-sided closed curve crossing no ai and bi , and that ai and bi are 2-sided closed curves crossing each other exactly once transversely and no other aj and bj with i = j . See the left-hand of Fig. 2. Then we write ρ = (ρ(x), ρ(a1 ), ρ(b1 ), . . . , ρ(ak ), ρ(bk )). For N2k , we express a cycle parity over N2k using a system {m, l, a1 , b1 , . . . , ak−1 , bk−1 } such that m and l are 1-sided and 2-sided closed curves, respectively, crossing at exactly one point, and {a1 , b1 , . . . , ak−1 , bk−1 } are the same as for N2k+1 . See the right-hand of Fig. 2. Then we write ρ = (ρ(m), ρ(l), ρ(a1 ), ρ(b1 ), . . . , ρ(ak−1 ), ρ(bk−1 )). Theorem 7 ([6]). Let ρ be a non-trivial cycle parity on the nonorientable closed surface Nk .
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Fig. 2. A set of generators for Nk
(i) If k ≥ 3 is odd, then ρ is congruent to either of type A = (1, 0, 0, . . . , 0), B = (1, 1, 0, . . . , 0) or C = (0, 1, 0, . . . , 0). If k = 1, then ρ is congruent to type A. (ii) If k ≥ 4 is even, then ρ is congruent to either of type D = (0, 0, 1, 0, . . . , 0), E = (0, 1, 0, 0, . . . , 0) or F = (1, 0, 0, 0, . . . , 0). If k = 2, then ρ is congruent to either of type E = (0, 1) or F = (1, 0). Let G be an even embedding on a closed surface F 2 . It is easy to see that any two homotopic closed walks of G have the same parity of length. Thus, the cycle parity ρG of G can be defined by ρG ([W ]) ≡ |W | (mod 2) for each closed walk W of G, where [W ] stands for the homotopy class of W and |W | the length of W . We call this ρG the cycle parity of G. Two cycle parities ρG and ρG of two even embeddings G and G are said to be congruent if there is a homeomorphism h : F 2 → F 2 with ρh(G) = ρG . It is obvious that an even embedding G is bipartite if and only if ρG is trivial, i.e., ρG ([W ]) ≡ 0 (mod 2) for all closed walk W of G. Let G be a non-bipartite even embedding on the orientable closed surface Sg with g > 0. Theorem 6 asserts that ρh(G) = (1, 0, . . . , 0) for some homeomorphism h : Sg → Sg , where ρh(G) = (ρ(a1 ), ρ(b1 ), . . . , ρ(ag ), ρ(bg )). Observe that the essential closed curve b1 is bipartizing in h(G) since all odd cycles of G are destroyed by the cutting. For nonorientable closed surfaces Nk , we can find such a bipartizing curve in even embeddings with cycle parities of type C, D and E. For even embeddings on Nk with other non-trivial cycle parities, we can find a simple closed curve which is assigned “1” and cuts open Nk into an orientable surface. The following theorem is a characterization for chromatic numbers of quadrangulations, observing the congruence classes of cycle parities as above. Theorem 8 ([6]). For any nonorientable closed surface Nk , there exists a positive integer R(k) such that any quadrangulation G on Nk with r(G) ≥ R(N ) is 4-colorable and: (i) G is 2-chromatic if and only if ρG is trivial. (ii) The following statements are equivalent. (a) G is 3-chromatic, (b) ρG is of either type C, D or E, (c) G is non-bipartite and has a bipartizing independent curve. (iii) The following statements are equivalent. (a) G is 4-chromatic,
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(b) ρG is of either type A, B or F , (c) G has an odd cycle cutting open Nk into an orientable surface. 3. Proof of the Theorem In this section, we shall prove Theorem 5. Theorem 8 asserts that Theorem 5 is true for quadrangulations on nonorientable closed surfaces. Hence we have to extend the argument to all even embeddings on all closed surfaces. We begin with proving the sufficiency. Proposition 9. If an embedding G on a closed surface F 2 admits a bipartizing independent curve, then G is 3-colorable. Proof. Suppose that G has a bipartizing independent curve l. By the definition, l intersects G only at their vertices. Let S be the set of vertices of G lying on l. Since G − S is bipartite and S is independent, we can construct a 3-coloring of G by giving a third color to S in addition to a 2-coloring of G − S. We use the following lemmas to take required cycles of a fixed homotopy type. Lemma 10 ([6]). Let ρ˜ be a fixed cycle parity over a closed surface F 2 and let be a fixed simple closed curve on F 2 . Let n be a natural number. Then, there exists a positive integer L(ρ, ˜ , n) such that for any quadrangulation H on F 2 with ρH ≡ ρ˜ and r(H ) ≥ L(ρ, ˜ , n), there exists a homeomorphism h on F 2 such that ρh(H ) = ρ˜ and h(H ) has n cycles homotopic to on F 2 . In particular, if is 2-sided and n ≥ 2, then these n cycles can be taken to be disjoint. We begin with the orientable case. Proposition 11. For any orientable closed surface Sg with g > 0, there exists an integer N(g) such that every 3-colorable even embedding on Sg with r(G) ≥ N (g) has a bipartizing independent curve. Proof. Let N(g) = max{M1 (g), L((1, 0, . . . , 0), b2 , 3)}, where M1 (·) and L(·, ·, ·) are the numbers in Theorem 2 and Lemma 10. Let G be any 3-colorable even embedding on Sg with r(G) ≥ N (g). Since r(G) ≥ M1 (g), we can simply assume that G is non-bipartite, by Theorem 2. Since r(G) ≥ L((1, 0, . . . , 0), b1 , 3), there exists a homeomorphism h on Sg such that (i) ρh(G) = (1, 0, . . . , 0), and (ii) h(G) has three disjoint homotopic even cycles C1 , C2 and C3 homotopic to b1 . Note that the graph obtained from G by cutting along C1 is bipartite. We may suppose that C1 , C2 and C3 lie on Sg in this order and denote by Gi the graph bounded by Ci and Ci+1 on an annulus, for i = 1, 2. Note that each Ci has even length since
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Fig. 3. How to take l1
it is homotopic to b1 and since ρh(G) (b1 ) ≡ 0 (mod 2). Therefore, Gi is bipartite for i = 1, 2. Now we shall take a bipartizing independent curve for G in G1 . Let C1 = b1 w1 · · · bm wm for some m ≥ 2, and let b1 , b11 , . . . , b1s , b2 be the neighbors of w1 in G1 lying in this order. Take a simple curve l1 starting at b1 passing through b11 , . . . , b1s in this order and ending at b2 . (See Fig. 3.) Similarly, take a simple curve li from bi to bi+1 through the neighbours of wi in G1 , for i = 2, . . . , k. Let = l1 ∪ · · · lk . Take a shortest simple closed curve in homotopic to C1 . Since is homotopic to C1 , is bipartizing for G. Let S be the vertices of G through which passes. Since G1 is bipartite, there is no edge joining two vertices of S in G1 . Moreover, there is no edge outside G1 joining two vertices of S lying on the same Ci , since G has no odd cycle bounding a disk on Sg . Furthermore, there is no edge outside G1 joining a vertex on C1 and a vertex on C2 , by the obstruction of C3 . Therefore, is independent in G. Hence is a bipartizing independent curve for G. For nonorientable closed surfaces, we first consider the projective-planar case with an additional restriction. Proposition 12. Let G be a non-bipartite even embedding on the projective plane which does not include a quadrangulation as a subgraph, and let v be a vertex of G. Then G admits a bipartizing independent curve not passing through any face incident to v. Proof. Since G has no quadrangulation as a subgraph, there exists a non-quadrilateral face f not surrounded by a 4-cycle. Let v1 · · · vm be the boundary cycle of f . If m ≥ 8, then we put a new vertex (dealt as v6 later) and join it to v5 , v7 , . . . , v1 . Moreover, if G has a non-quadrangular face f besides f , then put a new vertex vf in the interior of f and join vf to every second vertices on the boundary cycle of f . It is obvious that if the resulting graph has a bipartizing independent curve, then so does G. Hence we may suppose that G has only one hexagonal face f with boundary cycle v1 v2 v3 v4 v5 v6 and all other faces are quadrilateral. Let H be the triangulation obtained from G by adding a single vertex into each face of G except f , and joining it to all vertices on the corresponding boundary. Since every cycle surrounding f has length at least 6 in G, so does it in H . Therefore, for any simple closed curve γ surrounding f , we have |H ∩ γ | ≥ 6. Hence we can apply the theorem by Robertson and Seymour in [7] (that is, if we let K be a
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Fig. 4. The strip B1
projective-planar embedding with a face f bounded by a 2m-cycle v1 · · · v2m , then G has m disjoint paths joining vi and vm+i , for i = 1, . . . , m, unless K admits a closed curve surrounding f meeting K at most 2m − 1 times). Hence, since G has no 4-cycle surrounding f , H has three pairwise disjoint paths Pi joining vi and vi+3 for i = 1, 2, 3. Suppose that P1 , P2 and P3 are chosen so that |V (P1 ∪ P2 ∪ P3 )| is mini¨ mum. Cutting open the Mobius band N1 − f along P1 , P2 and P3 , we obtain three strips. We denote the three strips bounded by {P1 , P2 }, {P2 , P3 } and {P3 , P1 } by B1 , B2 and B3 , respectively. (See Fig. 4, in which identify each pair of antipodal points on the boundary of the disk.) Let Gi be the subgraph of G in Bi , for i = 1, 2, 3. We may suppose that the specified vertex v is contained in G3 − V (P1 ). (For otherwise, exchange the indices of Bi ’s.) Clearly, Gi is bipartite, for i = 1, 2, 3. We claim that in a proper 2-coloring of G1 , v1 and v4 have the same color. (If v1 and v5 have the same color, then the 2-coloring of G1 extends to that of G, and hence this contradicts that G is non-bipartite.) Observe that each Pi might have a vertex not in G, that is, a vertex of degree 4, say x, in V (H ) − V (G). By the minimality of |V (P1 ∪ P2 ∪ P3 )|, if such x is contained in some Pi , then Pi passes through ei and ei+2 for some i among the four edges e0 , e1 , e2 , e3 incident to x in this cyclic order around x in H , where the subscripts are taken modulo 4. Let G1 be the graph obtained from G1 by adding each vertex x ∈ V (P1 ∪ P2 ) − V (G) as a vertex of degree 2 adjacent to the two vertices to which x is adjacent in either P1 or P2 . Then G1 is clearly bipartite. We shall find a bipartizing independent curve for G. Since v1 and v4 have the same color (say black) in the 2-coloring of G1 , we can put P1 = b1 w1 · · · bm wm bm+1 for some m ≥ 1, where b1 = v1 and bm+1 = v4 . Now we proceed similarly to the proof of Proposition 11. First let l be a simple curve in f joining v1 and v4 . Take a simple curve li starting at bi and ending at bi+1 and passing through the neighbors of wi in G1 , for i = 1, . . . , m, and let be a shortest essential simple closed curve contained in l ∪ l1 ∪ · · · ∪ lm . We claim that l passes through no vertex x ∈ V (G1 ) − V (G1 ), since x has degree 2 in G1 and since it appears only on either
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P1 or P2 . Clearly, cuts open the projective plane N1 into a disk, and every even embedding on the disk is bipartite. Hence is bipartizing for G. Moreover, since v is contained in G3 − V (P1 ), does not pass through any face incident to v. Now we have only to prove that l is independent in G. Let S be the vertices of G through which passes. Observe that S is independent in G1 since S is a subset of the black vertices of G1 . If there are two adjacent vertices x, y ∈ S lying on the same Pi , then we can find an odd cycle bounding a disk, a contradiction. If there are two adjacent vertices x, y ∈ S such that x ∈ V (P1 ) and y ∈ V (P2 ), then we cannot take P3 , a contradiction. Therefore, is a bipartizing independent curve for G. Lemma 13. For any nonorientable closed surface Nk with k ≥ 2, there exists a positive integer L(k) satisfying the following: Let G be an even embedding on Nk with representativity at least L(k). If G has cycle parity of type A, B or F, then G has a cycle separating G into a non-bipartite even embedding on the M¨obius band and a bipartite even embedding on the remaining punctured surface. Proof. Take k disjoint 1-sided simple closed curves x1 , x2 , . . . , xk in Nk . The cycle parity ρ over Nk can also be expressed by ρ = [ρ(x1 ), . . . , ρ(xk )]. It was shown in [6] that B = [1, 0, . . . , 0] and F = [1, 0, . . . , 0]. Therefore, for cycle parities of type A = (1, 0, . . . , 0), B = [1, 0, . . . , 0] and F = [1, 0, . . . , 0], we can find a unique closed curve such that ρ() = 1 among the k closed curves used for expressing cycle parities. Moreover, is a 1-sided simple closed curve. Then the twice of , denoted ¨ by 2 , is a 2-sided simple closed curve separating a Mobius band containing x from Nk such that ρ(2 ) = ρ()ρ() = 0. By Lemma 10, H has a cycle homotopic to 2 , since we may suppose r(H ) ≥ R(ρ, 2 , 1) for ρ of type A, B and F. Proposition 14. The necessity of Theorem 5 holds for all nonorientable closed surfaces. In particular, no restriction for the representativity is needed in the projective planar case. Proof. Let N(k) = max{R(k), L(k)}, where R(·) and L(·) are the numbers in Theorem 8 and Lemma 13. Let G be a 3-colorable even embedding on Nk with representativity at least N (k). We may suppose that χ (G) = 3. For each non-quadrilateral face of G, put a new vertex and join it to every second vertices on the corresponding boundary cycle. The resulting graph is a quadrangulation including G, which is called a quadrangular extension of G and denoted by ˜ Note that ρG = ρ ˜ . G. G ˜ = 4, by Theorem 3. Since G is 3-chroFor the projective plane N1 , we have χ (G) matic, G includes no quadrangulation as a subgraph. Therefore, by Proposition 12, G admits a bipartizing independent curve. Note that we need no restriction for the representativity. ˜ − Now consider the nonorientable closed surface Nk with k ≥ 2. Let S = V (G) ˜ ˜ ˜ V (G). By Theorem 8, we have χ (G) ≤ 4 since r(G) ≥ R(k). If χ (G) = 3, then ρG˜ ˜ has a bipartizig independent curve l. is of type C, D or E. Then, by Theorem 8, G ˜ we can find a bipartizing independent curve for G. Hence we Removing S from G,
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˜ is 4-chromatic. Then the cycle parity of G ˜ is of type A, B or F, by Thesuppose that G orem 8. Since G is 3-colorable and since ρG˜ = ρG , G includes no quadrangulation as a subgraph, by Theorem 8. We shall prove that G admits a bipartizing independent curve. Similarly to the proof of Proposition 12, we may suppose that G has only one hexagonal face f not surrounded by a 4-cycle and all other faces of G are quadrilateral. By Lemma 13, since r(G) ≥ L(k), G has a cycle C separating G into a non-bipartite quadrangula¨ tion on the Mobius band, denoted by G1 , and a bipartite quadrangulation on the remaining punctured surface, denoted by G2 , where G1 ∩ G2 = C. Note that C can be chosen in G so that f is a face of G1 , since G has sufficiently large representativity. ˜ 1 be an even embedding on N1 obtained from G by pasting a disk to the Let G ¨ boundary component of the Mobius band and adding a new vertex v1 in the disk to ˜ 1 is a non-bipartite quadrangulation join it to every second vertices on C. Then G ˜ 1 has a bipartizing independent on the projective plane. Hence, by Proposition 12, G curve l not passing through any face incident to v1 . Since l can be taken in G1 and G2 is bipartite, corresponds to a bipartizing independent curve for G. Proof of Theorem 5. Theorem 5 follows from Propositions 11 and 14.
Finally, we pose a further problem. As in Proposition 14, Theorem 5 holds for the projective plane with no restriction for the representativity. Moreover, quadrangulations on the torus and the Klein bottle don’t require those restrictions, either [1]. Problem 15. Can the condition for the representativity be omitted from Theorem 5? References 1. Archdeacon, D., Hutchinson, J.P., Nakamoto, A., Negami, S., Ota, K.: Chromatic Numbers of Quadrangulations on Closed Surfaces. J. Graph Theory 37, 100–114 (2001) 2. Hutchinson, J.P.: On coloring maps made from Eulerian graphs. Proceeding of 5th British Combinatorial Conference, 1975”, pp. 343–354 3. Hutchinson, J.P.: Three-coloring graphs embedded on surfaces with all faces even-sided. J. Combin. Theory, Ser. B 65 139–155 (1995) 4. Mohar, B., Seymour, P.D.: Coloring locally bipartite graphs on surfaces. J. Combin. Theory, Ser. B 84, 301–310 (2002) 5. Negami, S., Nakamoto, A.: Diagonal transformations of graphs on closed surfaces. Sci. Rep. Yokohama Nat. Univ., Sec. I 40, 71–97 (1993) 6. Nakamoto, A., Negami, S., Ota, K.: Chromatic Numbers and Cycle Parities of Quanragulations on Nonorientable Closed Surfaces. Discrete Math. 285, 211–218 (2004) 7. Robertson, N., Seymour, P.D.: Graph Minors. VII, Disjoint paths on a surface. J. Combin. Theory, Ser. B 45, 212–254 (1988) 8. Youngs, D.A.: 4-chromatic projective graphs. J. Graph Theory 21, 219–227 (1996)
Received: June 14, 2005 Final Version received: June 17, 2006
