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N -Flips in even triangulations on the projective plane
Discrete Mathematics 308 (2008) 5454–5462 www.elsevier.com/locate/disc
N -flips in even triangulations on the projective plane Atsuhiro Nakamoto a , Yusuke Suzuki b a Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-ku,
Yokohama 240-8501, Japan b Tsuruoka National College of Technology, Tsuruoka, Yamagata 997-8511, Japan
Received 29 August 2006; received in revised form 3 October 2007; accepted 3 October 2007 Available online 26 November 2007
Abstract We shall show that any two even triangulations G and G 0 on the projective plane with |V (G)| = |V (G 0 )| ≥ 14 can be transformed into each other by two operations called an N -flip and P2 -flip, if and only if both of them are simultaneously 3colorable, or not. c 2007 Elsevier B.V. All rights reserved.
Keywords: Even triangulation; N -flip; Projective plane
1. Introduction A triangulation G of a closed surface F 2 is a simple graph (with no loops and no multiple edges) embedded on the surface so that each face is triangular. A triangulation G is said to be even if each vertex of G has even degree. We don’t regard K 3 on the sphere as a triangulation, and hence we can say that the smallest even triangulation on the sphere is the octahedron, which is the complete tripartite graph K 2,2,2 as an abstract graph. Even triangulations are sometimes called Eulerian triangulations, for example, in [1,2]. A simple closed curve l on a closed surface F 2 is trivial if l bounds a 2-cell on F 2 ; otherwise l is essential. We apply these definitions to cycles of graphs embedded in the surface, regarding them as simple closed curves. (The readers should refer for other definitions to [3].) It is an important property of the projective plane that any two essential simple closed curves are homotopic, and they have an odd number of crossings. Assume that an even triangulation G has hexagonal region v1 v2 v3 v4 v5 v6 with diagonals v1 v3 , v3 v6 and v4 v6 and no inner vertices. The N -flip of the path v1 v3 v6 v4 is the operation of replacing the diagonals v1 v3 , v3 v6 and v4 v6 with v1 v5 , v2 v5 and v2 v4 in the hexagonal region. (See the top of Fig. 1.) We don’t apply an N -flip if the graph obtained from G by it is not simple. An N -flip obviously transforms an even triangulation into an even triangulation. Two even triangulations G and G 0 are said to be N -equivalent and denoted by G ∼ N G 0 if G and G 0 can be transformed into each other by a sequence of N -flips. Note that we deal with unlabeled triangulations. Let G be a 3-colorable even triangulation. Then V (G) can uniquely be decomposed into V R (G) ∪ VB (G) ∪ VY (G), where these classes are referred to as red, blue, and yellow vertices of G, respectively. Such a decomposition of V (G) E-mail addresses: [email protected] (A. Nakamoto), [email protected] (Y. Suzuki). c 2007 Elsevier B.V. All rights reserved. 0012-365X/$ - see front matter doi:10.1016/j.disc.2007.10.015
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Fig. 1. N -flips.
is called a tripartition of G. Observe that if an even triangulation is 3-colorable, then the N -flip clearly preserves the 3-colorability. Moreover, an N -flip preserves the tripartition of G. Nakamoto et al. has proved the following theorem in [4]: Theorem 1. Two even triangulations G and G 0 on the sphere are N -equivalent if |V R (G)| = |V R (G 0 )|, |VB (G)| = |VB (G 0 )| and |VY (G)| = |VY (G 0 )|. In this paper, we deal with even triangulations on the projective plane. Though the projective plane admits non3-colorable even triangulations, we first consider 3-colorable triangulations. The following is our first result in this paper. Theorem 2. Two 3-colored even triangulations G and G 0 on the projective plane are N -equivalent if |V R (G)| = |V R (G 0 )|, |VB (G)| = |VB (G 0 )| and |VY (G)| = |VY (G 0 )|. Let D be an even embedding on a closed surface F 2 , that is, a graph on F 2 such that each face is bounded by a cycle of even length. The face subdivision of D, denoted by S(D), is the graph obtained from D by adding a new vertex into each face of D and joining it to all vertices on the corresponding boundary cycle. The set V (S(D))− V (D) is called the color factor of S(D). Clearly, S(D) is an even triangulation. Theorem 3 (Mohar [2]). Every even triangulation G on the projective plane is a face subdivision of some even embedding, and the color factor of G can uniquely be taken if G is non-3-colorable. Let G be an even triangulation on the projective plane. If G is 3-colorable, then there are three ways to take a color factor of G, since each partite set of V (G) can be taken as in the color factor of G. On the other hand, if G is not 3-colorable, then G has a unique color factor, by Theorem 3. We denote the unique color factor of G by U (G) throughout the paper. It is easy to see that an N -flip in a non-3-colorable even triangulation does not change its color factor. (See Fig. 1. Both of the color factors of the upper and lower graphs are expressed as white squares.) The following is our second result for non-3-colorable even triangulations. Theorem 4. Two non 3-colorable even triangulations G and G 0 on the projective plane are N -equivalent if |V (G)| = |V (G 0 )| and |U (G)| = |U (G 0 )|. We consider another operation called a “P2 -flip”. Let G be an even triangulation on a closed surface and let v be a vertex of G with its link v1 · · · vk . Put two vertices x and y on vv1 and join them to v2 and vk , and let G 0 be the resulting graph. The P2 -flip of {x, y} is the operation of moving the inserted vertices x and y to the edge vv2 and joining them to v1 and v3 , as shown in Fig. 2. This operation always preserves the simpleness of the graph since G is simple. Clearly, if G is 3-colored, a P2 -flip changes its tripartition. On the other hand, if the graph is not 3-colorable, the operation changes its color factor. Using P2 -flips in addition to N -flips, we can change the size of tripartite sets or the color factor of even triangulations. The following are our second pair of theorems in the paper.
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Fig. 2. P2 -flip.
Fig. 3. Three local deformations for even triangulations.
Theorem 5. Two 3-colored even triangulations on the projective plane with the same number of vertices can be transformed into each other by a sequence of N -flips and P2 -flips if and only if exactly one of them is not isomorphic to I5 shown in Fig. 4. Theorem 6. Two non-3-colorable even triangulations on the projective plane with the same number of vertices can be transformed into each other by a sequence of N -flips and P2 -flips if and only if they are not isomorphic to I7 and I10 shown in Fig. 4. Since each of I5 , I7 and I10 has at most thirteen vertices, and since both an N -flip and a P2 -flip preserve the 3-colorability of the graphs, we have the following corollary from Theorems 5 and 6. Corollary 7. Any two even triangulations G and G 0 on the projective plane with |V (G)| = |V (G 0 )| ≥ 14 can be transformed into each other by a sequence of N -flips and P2 -flips if and only if they are simultaneously 3-colorable or not. 2. N-flips and irreducible even triangulations Let G be an even triangulation and let v be its vertex of degree 4 with link v1 v2 v3 v4 . The contraction of v at {v2 , v4 } is the operation of eliminating the vertex v, identifying two vertices v2 and v4 , and replacing the resulting two pairs of double edges by two single edges, as shown in the top of Fig. 3. If this operation yields multiple edges or loops, then we don’t apply it. Note that we define the contraction only for a vertex of degree 4, and that the contraction transforms an even triangulation G into an even triangulation with |V (G)| − 2 vertices. For simple notation, we call a contraction of a vertex of degree 4 a 4-contraction. Moreover, it is easy to see that this operation preserves the 3-colorability of graphs. Let G be an even triangulation and let f be a face of G bounded by v1 v2 v3 . The addition of an octahedron to f is the operation of adding inside of f a 3-cycle u 1 u 2 u 3 and edges u i v j for each distinct i, j ∈ {1, 2, 3}. Sometimes, the graph induced by {v1 , v2 , v3 , u 1 , u 2 , u 3 } is called an octahedral part of a graph. Furthermore, we call the inverse operation of the addition of an octahedron the removal of an octahedron. (See the center of Fig. 3.) Let G be an even triangulation and let u 1 and u 2 be two adjacent vertices of degree 4 with links v1 v2 u 2 v4 and u 1 v2 v3 v4 , respectively.
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Fig. 4. Irreducible even triangulations of the projective plane.
Fig. 5. Inside a quadrilateral region.
The twin-contraction is the operation of eliminating two vertices u 1 and u 2 , identifying v2 and v4 , and replacing two pairs of double edges by single edges, respectively, as shown in the bottom of Fig. 3. As well as a 4-contraction, we don’t apply any twin-contraction if it results in a non-simple graph. Note that each of the above three local deformations preserves the 3-colorability. Let G be an even triangulation on the projective plane and let v be a vertex of degree 4 with link v1 v2 v3 v4 . We suppose that another vertex x is incident to v1 , v2 , v3 , v4 so that both 3-cycles xv1 v2 and xv3 v4 are essential, and that 3-cycles xv1 v4 and xv2 v3 bound faces. Let D be the 2-cell region of G bounded by xv1 v2 xv3 v4 and containing v as an unique inner vertex. We call D a blindfold and x a terminal. Note that we cannot apply a 4-contraction at v. An even triangulation G is said to be irreducible if G does not admit a 4-contraction, a removal of an octahedron or a twin-contraction. Suzuki and Watanabe have determined the complete list of irreducible even triangulations on the projective plane [5], as in Fig. 4, where each pair of antipodal points on each hexagon should be identified. Note that I9 [n] contains exactly three blindfolds in the shaded region, but we can naturally increase the number of the blindfolds to any n ≥ 3 so that the resulting graph admits none of the three local transformations. Hence the notation I9 [n] expresses the embedded graph with exactly n blindfolds in the shaded region of the figure. If n = 1, 2, these are isomorphic to I1 and I8 , respectively. Now we shall prove the following theorem. Theorem 8. Every even triangulation on the projective plane can be transformed into one of I1 , I3 , I5 , I7 , I8 and I10 in Fig. 4 by a sequence of 4-contractions and N -flips. For proving the above theorem, we use the following lemmas. Lemma 9 (Suzuki and Watanabe [5]). Let G be an irreducible even triangulation on any closed surface F 2 except the sphere. Then the interior of a trivial 4-cycle in G has one of the structures shown in Fig. 5. Lemma 10. Let G be an even triangulation on a closed surface F 2 . Let G 1 and G 2 be even triangulations obtained from G by an addition of an octahedron. Then, G 1 and G 2 are N -equivalent.
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Fig. 6. Moving an octahedron.
Fig. 7. Reduction of the number of vertices of I8 + O.
Proof. Suppose that G is obtained from G 1 by removing three vertices u 1 , u 2 and u 3 forming a facial 3-cycle from a triangular region bounded by v1 v2 v3 , where u i v j ∈ E(G) for any distinct i, j ∈ {1, 2, 3}. Let v2 u 3 u 2 v3 w1 w2 · · · w2k be the link of v1 in G 1 . Note that we have k ≥ 1. Let G 2 be another even triangulation obtained from G by a single addition of an octahedron to the face bounded by a 3-cycle v1 v2 w2k . To prove the lemma, it suffices to show that G 1 and G 2 are N -equivalent, since G is a connected graph. First, we apply an N -flip of the path u 2 v3 v1 w1 in G 1 and let G 01 be the resulting graph. (See the first N -flip in Fig. 6.) Since for any i and j, u i and w j are separated by the 3-cycle v1 v2 v3 in G 1 , they are not adjacent in G 1 . Moreover, v2 , v3 , w1 , . . . , w2k are all distinct since they lie on the link of v1 in G 1 . Hence G 01 is simple. If k = 1, then G 01 is the graph that we required. Therefore, we assume that k ≥ 2. Now apply the second N -flip of the path u 2 w2 v1 w3 in G 01 . (See the second N -flip in Fig. 6.) It is easy to see that k N -flips can move an octahedron to a neighboring face in G. Let G + O denote an even triangulation on the projective plane obtained from G by an addition of an octahedron. The above lemma guarantees that G + O represents a unique even triangulation, up to N -equivalence. Lemma 11. G + O can be reduced by using N -flips and 4-contractions unless G is isomorphic to I1 . Proof. Let u 1 , u 2 and u 3 be vertices forming a facial 3-cycle in a triangular region bounded by v1 v2 v3 in G + O, like in the proof of the above lemma. Furthermore, let v2 u 3 u 2 v3 w1 w2 · · · w2k be the link of v1 . First, suppose that k ≥ 3. Apply the N -flip of the path u 2 v3 v1 w1 in G + O and let G 0 be the resulting graph. We assume that v2 w2 ∈ E(G + O); otherwise we could apply a 4-contraction of u 2 at {v2 , u 2 }. Since an N -flip of the path u 2 w2 v1 w3 in G 0 makes the link of u 2 into v1 u 3 u 1 w4 , we also assume that v2 w4 ∈ E(G + O). Similarly, we may assume that v2 w2i ∈ E(G + O) for i = 1, . . . , k − 1. Now, apply the sequence of N -flips of the path u 3 v2 v1 w2k and u 3 w2k−1 v1 w2k−2 in the graph G + O. We can apply a 4-contraction of u 2 at {u 3 , v3 } in the resulting graph. Otherwise, there would exist v3 w2k−3 ∈ E(G + O), a contradiction to G being projective-planar. If G has a vertex of degree at least 8, the above argument works by Lemma 10. Because the projective plane admits no 4-regular triangulation, G includes a vertex v = v1 of degree 6. Let v2 v3 w1 w2 w3 w4 be the link of v1 and assume that a facial 3-cycle u 1 u 2 u 3 is in the face of G bounded by v1 v2 v3 , where u i v j ∈ E(G) (i 6= j). Like for the case k ≥ 3, there must be v2 w2 , v3 w3 ∈ E(G + O). By symmetry, we have w1 w4 ∈ E(G + O). Under this condition, we find three quadrangular regions bounded by v2 v3 w3 w2 , v3 w1 w4 w3 and w1 w2 v2 w4 , respectively. Considering the degree conditions of the whole graph and Lemma 9, we have the following two cases. Case (a). Each of the three quadrangular regions has only one inner vertex. Obviously, G is equivalent to I8 . See Fig. 7, in which we can reduce the number of vertices.
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Fig. 8. I 0 and an N -flip in I9 .
Case (b). Each of the three quadrangular regions has only one diagonal. Suppose that v2 w3 , w1 w3 , w1 v2 ∈ E(G + O), by symmetry. In this case, G is nothing but I1 . Note that the right-hand example of Fig. 5 does not occur because it includes a contractible vertex of degree 4. Lemma 12. Let G be an even triangulation on the projective plane. Let G 0 be an even triangulation which is transformed into G by a single twin-contraction. Then G 0 can be reduced by using N -flips and 4-contractions unless G is the graph shown in the left-hand example of Fig. 8. Proof. It is easy to see that G 0 can be transformed into some even triangulation with an octahedron added by a sequence of N -flips. Hence, by Lemma 11, unless G 0 is N -equivalent to I1 + O, the number of vertices of G 0 can be reduced. Clearly, the exception of the lemma is a unique even triangulation which is transformed into I1 + O by a single twin-contraction. Lemma 13. The even triangulation I9 [n] with n ≥ 3 can be reduced by using N -flips and 4-contractions. Proof. See the N -flip in the center and the on right-hand side in Fig. 8. Only one N -flip yields an even triangulation that has a twin-contractible part. Now, we have prepared everything for proving Theorem 8. Proof of Theorem 8. By Lemmas 11–13 and [5], every even triangulation on the projective plane can be transformed into one of I1 , I2 , I3 , I4 , I5 , I6 , I7 , I8 , I10 , I11 , I1 + O and I10 by a sequence of N -flips and 4-contractions, where I10 is the one transformed into I1 by a single twin-contraction. According to the proof of Lemma 11, it is clear that I1 + O and I10 are N -equivalent. Moreover, each of them is N -equivalent to I10 . (See the top of Fig. 9.) Next we show that I2 and I4 can be reduced. It is easy to see that the resulting graphs in the figures include such vertices of degree 4. Two remaining ones in Fig. 9 present the facts that I6 ∼ N I8 + O and that the number of vertices of I11 can be reduced. 3. 2-subdivisions of edges Let G be an even triangulation on a closed surface F 2 and let e = ab be an edge of G sharing two triangular faces abc and abd. Put two vertices u, v on the edge ab so that a, u, v, b lie in this order, and join u and v to c and d. We call this operation a 2-subdivision, and {u, v} a 2-subdividing pair. Lemma 14 ([4]). Let K be an even triangulation on any surface F 2 , and let e1 , . . . , e2k be the edges incident to a vertex v of K appearing in the cyclic order around v, where deg(v) = 2k. Let G (resp., G 0 ) be the even triangulation on F 2 obtained from K by a 2-subdivision of ei (resp., ei+2 ). Then G and G 0 are N -equivalent. Lemma 15 ([4]). Let K be an even triangulation on any surface F 2 , and let v be a vertex v of degree 4 with link v1 · · · v4 . Suppose that the link of v1 in K is v2 vv4 x1 · · · x2h−3 , where deg(v1 ) = 2h ≥ 4. Let K 0 be the graph obtained from K by a 4-contraction of v at {v1 , v3 }. Then K can be transformed into the graph obtained from K 0 by adding a 2-subdividing pair to the edge x1 [v1 v3 ], where [v1 v3 ] is the vertex arising from the identification of v1 and v3 . We may suppose that the vertices of a 3-colored even triangulation are colored red, blue and yellow and denote the three color classes by V R (G), VB (G) and VY (G), respectively. We say that an edge with red and blue endpoints is an r b-edge. Similarly, we can define a by-edge and an r y-edge.
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Fig. 9. N -flips in graphs.
Observe that if ei in Lemma 14 is an r b-edge, then so is ei+2 . (Other cases are similar.) Therefore, we can denote an even triangulation obtained from G by adding p, q and r 2-subdividing pairs to an r b-edge, a by-edge and an r y-edge, respectively, by G + r b( p) + by(q) + r y(r ). This notation might express various even triangulations, but they are all N -equivalent, by Lemma 14. Now consider the case when G is non-3-colorable. Recall that a non-3-colorable even triangulation G on the projective plane admits a unique color factor U such that G − U is a non-bipartite even embedding. Let Fr (G) denote the set of edges of G − U , and let Sp(G) = E(G) − Fr (G). If e ∈ Sp(G), we call e an sp-edge, while e ∈ Fr (G) is an f r -edge. Like for the 3-colored case, if ei is an f r -edge (or an sp-edge) in Lemma 14, then so is ei+2 . (In particular, if ei is an sp-edge and so is ei+1 , we can also move the 2-subdividing pair to ei+1 ; otherwise, G − U would be a bipartite even embedding.) Therefore, we can denote an even triangulation obtained from G by adding p and q 2-subdividing pairs to an sp-edge and an f r -edge, respectively, by G + sp( p) + f r (q). It has precisely |U | + p independent vertices and |X | + p + 2q other vertices, and hence has |V (G)| + 2 p + 2q vertices in total. Applying Lemmas 14 and 15, we obtain the following lemma. Lemma 16. Let G be an even triangulation on the projective plane. Suppose that G can be transformed into G 0 by a sequence of 4-contractions and N -flips. (i) If G is 3-colored, then G ∼ N G 0 + r b( p) + by(q) + r y(r ), where |V R (G)| − |V R (G 0 )| = α,
|VB (G)| − |VB (G 0 )| = β,
|VY (G)| − |VY (G 0 )| = γ
and p + r = α, p + q = β, q + r = γ . (ii) If G is not 3-colorable, then G ∼ N G 0 + sp( p) + f r (q), where |U (G)| − |U (G 0 )| = p and |V (G)| − |V (G 0 )| = p + 2q. In Lemma 16, note that if G and G 0 are given, then p, q, r can be determined uniquely.
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Fig. 10. N -flips in I5 + by(1).
4. Proof of Theorems 2 and 4–6 Recall that a 4-contraction reduces the number of vertices by 2, and hence it does not change the parity of the number of vertices. Moreover, an N -flip and a 4-contraction preserve 3-colorability. Therefore, we classify even triangulations on the projective plane into four classes, depending on the 3-colorability and the parity of the number of vertices. By Theorem 8, we have six minimal even triangulations in the projective plane with respect to N -flips and 4contractions. Among them, only I8 is a 3-colorable even triangulation with an even number of vertices. That is, every 3-colorable even triangulation on the projective plane with an even number of vertices can be transformed into I8 by a sequence of N -flips and 4-contractions. Similarly, I3 and I8 are 3-colorable with odd number of vertices. The remaining ones I1 , I7 and I10 are not 3-colorable and have 7, 10 and 10 vertices, respectively. Proof of Theorem 2. Let G be any 3-colored even triangulation on the projective plane. Then, by Theorem 8, there exists a sequence of even triangulations G = G 0 , G 1 , . . . , G k such that G i+1 is obtained from G i by a single N -flip or a single 4-contraction, for i = 0, . . . , k − 1, and G k is one of I3 , I5 and I8 . We first consider the case when |V (G)| is even. Since G k must be isomorphic to I8 , we have G ∼ N I8 + r b( p) + by(q) + r y(r ), where |V R (G)| = p + r + 3, |VB (G)| = p + q + 3 and |VY (G)| = q + r + 4, by Lemma 16. Then this obviously proves the theorem. Now we prove the theorem when |V (G)| is odd. In this case, note that G k is isomorphic to either I3 or I5 . Hence if both G and G 0 can be transformed into exactly one of I3 and I5 by 4-contractions and N -flips, then the same argument as above follows. Let G be a 3-colored even triangulation which is transformed into I5 by 4-contractions and N -flips. Then we have |V R (G)| = p +r + 6, |VB (G)| = p + q + 4 and |VY (G)| = q +r + 3 for some integers p, q, r ≥ 0. It suffices to show that G ∼ N I3 + r b( p 0 ) + by(q 0 ) + r y(r 0 ) for some p 0 , q 0 , r 0 . Observe that I5 + by(1) ∼ N I3 + r b(2) + yr (1), which is shown in Fig. 10. Since G ∼ N I5 + r b( p) + by(q) + r y(r ), we have G ∼ N I3 + r b( p + 2) + by(q − 1) + r y(r + 1), if q ≥ 1. Hence we suppose that q = 0. Then we have |V R (G)| = p + r + 6, |VB (G)| = p + 4 and |VY (G)| = r + 3 for some p, r ≥ 0. Now we prove that there do not exist p 0 , q 0 , r 0 ≥ 0 such that G ∼ N I3 +r b( p 0 ) + by(q 0 ) +r y(r 0 ). For a contradiction, suppose that p 0 , q 0 , r 0 ≥ 0 exist. Since I3 has exactly three yellow vertices, and since |VY (G)| = r + 3, we have q 0 + r 0 = r . Since q 0 , r 0 ≥ 0, we have |q 0 − r 0 | ≤ r . On the other hand, since |V R (G)| − |VB (G)| = r + 2, we have r 0 − q 0 = r + 2, a contradiction. Proof of Theorem 5. It is easy to see that I8 + r b( p) + by(q) + r y(r ) with q ≥ 1 can be transformed into I8 + r b( p + 1) + by(q − 1) + r y(r ) by N - and P2 -flips. Thus, every 3-colored even triangulation on the projective plane with an even number of vertices can be transformed into I8 + r b(m), where |V (G)| = 2m + 10. Similarly, one with an odd number of vertices, except I5 , can be transformed into I3 +r b(m) where |V (G)| = 2m +9. The exception I5 clearly does not have two adjacent vertices of degree 4; thus we can’t apply a P2 -flip to it. Furthermore, we can see that I5 is the face subdivision of K 3,4 . This means that any N -flip to the graph would yield multiple edges. Therefore, we can’t transform I5 into any other graph by the two operations. Then, the theorem holds. Proof of Theorem 4. We use the same method as in the proof of Theorem 2. Let G be a non 3-colorable even triangulation on the projective plane. We first prove the case when |V (G)| is odd. In this case, G is transformed into I1 by a sequence of 4-contractions and N -flips. Therefore, we can prove that G ∼ N I1 + sp( p) + f r (q), where |U (G)| = p + 3 and |X (G)| = p + 2q + 4. When |V (G)| is even, G is transformed into either I7 or I10 by 4-contractions and N -flips, each of which has 10 vertices. If G and G 0 are transformed into one of I7 and I10 , then the same argument as in the proof
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Fig. 11. N -flips in I10 + f r (1).
of Theorem 2 follows. Hence we suppose not. Fig. 11 shows that I10 + f r (1) ∼ N I7 + sp(1). This implies that G ∼ N I10 + sp( p − 1) + f r (q) ∼ N I7 + sp( p) + f r (q − 1), if p, q ≥ 1. Clearly, if one of p = 0 and q = 0 holds, then each of G and G 0 can be transformed into exactly one of I7 and I10 , a contradiction. Proof of Theorem 6. We can prove the theorem by the same method as in the proof of Theorem 5. Consider the minimality of the graphs in Fig. 4; we can’t apply both an N - and a P2 -flip to I7 . References [1] [2] [3] [4] [5]
J.P. Hutchinson, B.R. Richter, P.D. Seymour, Colouring Eulerian triangulations, J. Combin. Theory Ser. B 84 (2002) 225–239. B. Mohar, Coloring Eulerian triangulations of the projective plane, Discrete Math. 244 (2002) 339–343. B. Mohar, C. Thomassen, Graphs on Surfaces, The Johns Hopkins University Press, 2001. A. Nakamoto, T. Sakuma, Y. Suzuki, N -flips in even triangulations on the sphere, J. Graph Theory 51 (2006) 260–268. Y. Suzuki, T. Watanabe, Generating even triangulations of the projective plane, J. Graph Theory 56 (2007) 333–349.
Further reading [1] J.P. Hutchinson, Three-coloring graphs embedded on surfaces with all faces even-sided, J. Combin. Theory Ser. B 65 (1995) 139–155. [2] A. Nakamoto, Quadrangulations on closed surfaces, Interdiscip. Inform. Sci. 7 (2001) 77–98. [3] S. Negami, Diagonal flips of triangulations on surfaces, a survey, Yokohama Math. J. 47 (1999) 1–40.
N -flips in even triangulations on the projective plane Atsuhiro Nakamoto a , Yusuke Suzuki b a Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-ku,
Yokohama 240-8501, Japan b Tsuruoka National College of Technology, Tsuruoka, Yamagata 997-8511, Japan
Received 29 August 2006; received in revised form 3 October 2007; accepted 3 October 2007 Available online 26 November 2007
Abstract We shall show that any two even triangulations G and G 0 on the projective plane with |V (G)| = |V (G 0 )| ≥ 14 can be transformed into each other by two operations called an N -flip and P2 -flip, if and only if both of them are simultaneously 3colorable, or not. c 2007 Elsevier B.V. All rights reserved.
Keywords: Even triangulation; N -flip; Projective plane
1. Introduction A triangulation G of a closed surface F 2 is a simple graph (with no loops and no multiple edges) embedded on the surface so that each face is triangular. A triangulation G is said to be even if each vertex of G has even degree. We don’t regard K 3 on the sphere as a triangulation, and hence we can say that the smallest even triangulation on the sphere is the octahedron, which is the complete tripartite graph K 2,2,2 as an abstract graph. Even triangulations are sometimes called Eulerian triangulations, for example, in [1,2]. A simple closed curve l on a closed surface F 2 is trivial if l bounds a 2-cell on F 2 ; otherwise l is essential. We apply these definitions to cycles of graphs embedded in the surface, regarding them as simple closed curves. (The readers should refer for other definitions to [3].) It is an important property of the projective plane that any two essential simple closed curves are homotopic, and they have an odd number of crossings. Assume that an even triangulation G has hexagonal region v1 v2 v3 v4 v5 v6 with diagonals v1 v3 , v3 v6 and v4 v6 and no inner vertices. The N -flip of the path v1 v3 v6 v4 is the operation of replacing the diagonals v1 v3 , v3 v6 and v4 v6 with v1 v5 , v2 v5 and v2 v4 in the hexagonal region. (See the top of Fig. 1.) We don’t apply an N -flip if the graph obtained from G by it is not simple. An N -flip obviously transforms an even triangulation into an even triangulation. Two even triangulations G and G 0 are said to be N -equivalent and denoted by G ∼ N G 0 if G and G 0 can be transformed into each other by a sequence of N -flips. Note that we deal with unlabeled triangulations. Let G be a 3-colorable even triangulation. Then V (G) can uniquely be decomposed into V R (G) ∪ VB (G) ∪ VY (G), where these classes are referred to as red, blue, and yellow vertices of G, respectively. Such a decomposition of V (G) E-mail addresses: [email protected] (A. Nakamoto), [email protected] (Y. Suzuki). c 2007 Elsevier B.V. All rights reserved. 0012-365X/$ - see front matter doi:10.1016/j.disc.2007.10.015
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Fig. 1. N -flips.
is called a tripartition of G. Observe that if an even triangulation is 3-colorable, then the N -flip clearly preserves the 3-colorability. Moreover, an N -flip preserves the tripartition of G. Nakamoto et al. has proved the following theorem in [4]: Theorem 1. Two even triangulations G and G 0 on the sphere are N -equivalent if |V R (G)| = |V R (G 0 )|, |VB (G)| = |VB (G 0 )| and |VY (G)| = |VY (G 0 )|. In this paper, we deal with even triangulations on the projective plane. Though the projective plane admits non3-colorable even triangulations, we first consider 3-colorable triangulations. The following is our first result in this paper. Theorem 2. Two 3-colored even triangulations G and G 0 on the projective plane are N -equivalent if |V R (G)| = |V R (G 0 )|, |VB (G)| = |VB (G 0 )| and |VY (G)| = |VY (G 0 )|. Let D be an even embedding on a closed surface F 2 , that is, a graph on F 2 such that each face is bounded by a cycle of even length. The face subdivision of D, denoted by S(D), is the graph obtained from D by adding a new vertex into each face of D and joining it to all vertices on the corresponding boundary cycle. The set V (S(D))− V (D) is called the color factor of S(D). Clearly, S(D) is an even triangulation. Theorem 3 (Mohar [2]). Every even triangulation G on the projective plane is a face subdivision of some even embedding, and the color factor of G can uniquely be taken if G is non-3-colorable. Let G be an even triangulation on the projective plane. If G is 3-colorable, then there are three ways to take a color factor of G, since each partite set of V (G) can be taken as in the color factor of G. On the other hand, if G is not 3-colorable, then G has a unique color factor, by Theorem 3. We denote the unique color factor of G by U (G) throughout the paper. It is easy to see that an N -flip in a non-3-colorable even triangulation does not change its color factor. (See Fig. 1. Both of the color factors of the upper and lower graphs are expressed as white squares.) The following is our second result for non-3-colorable even triangulations. Theorem 4. Two non 3-colorable even triangulations G and G 0 on the projective plane are N -equivalent if |V (G)| = |V (G 0 )| and |U (G)| = |U (G 0 )|. We consider another operation called a “P2 -flip”. Let G be an even triangulation on a closed surface and let v be a vertex of G with its link v1 · · · vk . Put two vertices x and y on vv1 and join them to v2 and vk , and let G 0 be the resulting graph. The P2 -flip of {x, y} is the operation of moving the inserted vertices x and y to the edge vv2 and joining them to v1 and v3 , as shown in Fig. 2. This operation always preserves the simpleness of the graph since G is simple. Clearly, if G is 3-colored, a P2 -flip changes its tripartition. On the other hand, if the graph is not 3-colorable, the operation changes its color factor. Using P2 -flips in addition to N -flips, we can change the size of tripartite sets or the color factor of even triangulations. The following are our second pair of theorems in the paper.
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Fig. 2. P2 -flip.
Fig. 3. Three local deformations for even triangulations.
Theorem 5. Two 3-colored even triangulations on the projective plane with the same number of vertices can be transformed into each other by a sequence of N -flips and P2 -flips if and only if exactly one of them is not isomorphic to I5 shown in Fig. 4. Theorem 6. Two non-3-colorable even triangulations on the projective plane with the same number of vertices can be transformed into each other by a sequence of N -flips and P2 -flips if and only if they are not isomorphic to I7 and I10 shown in Fig. 4. Since each of I5 , I7 and I10 has at most thirteen vertices, and since both an N -flip and a P2 -flip preserve the 3-colorability of the graphs, we have the following corollary from Theorems 5 and 6. Corollary 7. Any two even triangulations G and G 0 on the projective plane with |V (G)| = |V (G 0 )| ≥ 14 can be transformed into each other by a sequence of N -flips and P2 -flips if and only if they are simultaneously 3-colorable or not. 2. N-flips and irreducible even triangulations Let G be an even triangulation and let v be its vertex of degree 4 with link v1 v2 v3 v4 . The contraction of v at {v2 , v4 } is the operation of eliminating the vertex v, identifying two vertices v2 and v4 , and replacing the resulting two pairs of double edges by two single edges, as shown in the top of Fig. 3. If this operation yields multiple edges or loops, then we don’t apply it. Note that we define the contraction only for a vertex of degree 4, and that the contraction transforms an even triangulation G into an even triangulation with |V (G)| − 2 vertices. For simple notation, we call a contraction of a vertex of degree 4 a 4-contraction. Moreover, it is easy to see that this operation preserves the 3-colorability of graphs. Let G be an even triangulation and let f be a face of G bounded by v1 v2 v3 . The addition of an octahedron to f is the operation of adding inside of f a 3-cycle u 1 u 2 u 3 and edges u i v j for each distinct i, j ∈ {1, 2, 3}. Sometimes, the graph induced by {v1 , v2 , v3 , u 1 , u 2 , u 3 } is called an octahedral part of a graph. Furthermore, we call the inverse operation of the addition of an octahedron the removal of an octahedron. (See the center of Fig. 3.) Let G be an even triangulation and let u 1 and u 2 be two adjacent vertices of degree 4 with links v1 v2 u 2 v4 and u 1 v2 v3 v4 , respectively.
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Fig. 4. Irreducible even triangulations of the projective plane.
Fig. 5. Inside a quadrilateral region.
The twin-contraction is the operation of eliminating two vertices u 1 and u 2 , identifying v2 and v4 , and replacing two pairs of double edges by single edges, respectively, as shown in the bottom of Fig. 3. As well as a 4-contraction, we don’t apply any twin-contraction if it results in a non-simple graph. Note that each of the above three local deformations preserves the 3-colorability. Let G be an even triangulation on the projective plane and let v be a vertex of degree 4 with link v1 v2 v3 v4 . We suppose that another vertex x is incident to v1 , v2 , v3 , v4 so that both 3-cycles xv1 v2 and xv3 v4 are essential, and that 3-cycles xv1 v4 and xv2 v3 bound faces. Let D be the 2-cell region of G bounded by xv1 v2 xv3 v4 and containing v as an unique inner vertex. We call D a blindfold and x a terminal. Note that we cannot apply a 4-contraction at v. An even triangulation G is said to be irreducible if G does not admit a 4-contraction, a removal of an octahedron or a twin-contraction. Suzuki and Watanabe have determined the complete list of irreducible even triangulations on the projective plane [5], as in Fig. 4, where each pair of antipodal points on each hexagon should be identified. Note that I9 [n] contains exactly three blindfolds in the shaded region, but we can naturally increase the number of the blindfolds to any n ≥ 3 so that the resulting graph admits none of the three local transformations. Hence the notation I9 [n] expresses the embedded graph with exactly n blindfolds in the shaded region of the figure. If n = 1, 2, these are isomorphic to I1 and I8 , respectively. Now we shall prove the following theorem. Theorem 8. Every even triangulation on the projective plane can be transformed into one of I1 , I3 , I5 , I7 , I8 and I10 in Fig. 4 by a sequence of 4-contractions and N -flips. For proving the above theorem, we use the following lemmas. Lemma 9 (Suzuki and Watanabe [5]). Let G be an irreducible even triangulation on any closed surface F 2 except the sphere. Then the interior of a trivial 4-cycle in G has one of the structures shown in Fig. 5. Lemma 10. Let G be an even triangulation on a closed surface F 2 . Let G 1 and G 2 be even triangulations obtained from G by an addition of an octahedron. Then, G 1 and G 2 are N -equivalent.
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Fig. 6. Moving an octahedron.
Fig. 7. Reduction of the number of vertices of I8 + O.
Proof. Suppose that G is obtained from G 1 by removing three vertices u 1 , u 2 and u 3 forming a facial 3-cycle from a triangular region bounded by v1 v2 v3 , where u i v j ∈ E(G) for any distinct i, j ∈ {1, 2, 3}. Let v2 u 3 u 2 v3 w1 w2 · · · w2k be the link of v1 in G 1 . Note that we have k ≥ 1. Let G 2 be another even triangulation obtained from G by a single addition of an octahedron to the face bounded by a 3-cycle v1 v2 w2k . To prove the lemma, it suffices to show that G 1 and G 2 are N -equivalent, since G is a connected graph. First, we apply an N -flip of the path u 2 v3 v1 w1 in G 1 and let G 01 be the resulting graph. (See the first N -flip in Fig. 6.) Since for any i and j, u i and w j are separated by the 3-cycle v1 v2 v3 in G 1 , they are not adjacent in G 1 . Moreover, v2 , v3 , w1 , . . . , w2k are all distinct since they lie on the link of v1 in G 1 . Hence G 01 is simple. If k = 1, then G 01 is the graph that we required. Therefore, we assume that k ≥ 2. Now apply the second N -flip of the path u 2 w2 v1 w3 in G 01 . (See the second N -flip in Fig. 6.) It is easy to see that k N -flips can move an octahedron to a neighboring face in G. Let G + O denote an even triangulation on the projective plane obtained from G by an addition of an octahedron. The above lemma guarantees that G + O represents a unique even triangulation, up to N -equivalence. Lemma 11. G + O can be reduced by using N -flips and 4-contractions unless G is isomorphic to I1 . Proof. Let u 1 , u 2 and u 3 be vertices forming a facial 3-cycle in a triangular region bounded by v1 v2 v3 in G + O, like in the proof of the above lemma. Furthermore, let v2 u 3 u 2 v3 w1 w2 · · · w2k be the link of v1 . First, suppose that k ≥ 3. Apply the N -flip of the path u 2 v3 v1 w1 in G + O and let G 0 be the resulting graph. We assume that v2 w2 ∈ E(G + O); otherwise we could apply a 4-contraction of u 2 at {v2 , u 2 }. Since an N -flip of the path u 2 w2 v1 w3 in G 0 makes the link of u 2 into v1 u 3 u 1 w4 , we also assume that v2 w4 ∈ E(G + O). Similarly, we may assume that v2 w2i ∈ E(G + O) for i = 1, . . . , k − 1. Now, apply the sequence of N -flips of the path u 3 v2 v1 w2k and u 3 w2k−1 v1 w2k−2 in the graph G + O. We can apply a 4-contraction of u 2 at {u 3 , v3 } in the resulting graph. Otherwise, there would exist v3 w2k−3 ∈ E(G + O), a contradiction to G being projective-planar. If G has a vertex of degree at least 8, the above argument works by Lemma 10. Because the projective plane admits no 4-regular triangulation, G includes a vertex v = v1 of degree 6. Let v2 v3 w1 w2 w3 w4 be the link of v1 and assume that a facial 3-cycle u 1 u 2 u 3 is in the face of G bounded by v1 v2 v3 , where u i v j ∈ E(G) (i 6= j). Like for the case k ≥ 3, there must be v2 w2 , v3 w3 ∈ E(G + O). By symmetry, we have w1 w4 ∈ E(G + O). Under this condition, we find three quadrangular regions bounded by v2 v3 w3 w2 , v3 w1 w4 w3 and w1 w2 v2 w4 , respectively. Considering the degree conditions of the whole graph and Lemma 9, we have the following two cases. Case (a). Each of the three quadrangular regions has only one inner vertex. Obviously, G is equivalent to I8 . See Fig. 7, in which we can reduce the number of vertices.
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Fig. 8. I 0 and an N -flip in I9 .
Case (b). Each of the three quadrangular regions has only one diagonal. Suppose that v2 w3 , w1 w3 , w1 v2 ∈ E(G + O), by symmetry. In this case, G is nothing but I1 . Note that the right-hand example of Fig. 5 does not occur because it includes a contractible vertex of degree 4. Lemma 12. Let G be an even triangulation on the projective plane. Let G 0 be an even triangulation which is transformed into G by a single twin-contraction. Then G 0 can be reduced by using N -flips and 4-contractions unless G is the graph shown in the left-hand example of Fig. 8. Proof. It is easy to see that G 0 can be transformed into some even triangulation with an octahedron added by a sequence of N -flips. Hence, by Lemma 11, unless G 0 is N -equivalent to I1 + O, the number of vertices of G 0 can be reduced. Clearly, the exception of the lemma is a unique even triangulation which is transformed into I1 + O by a single twin-contraction. Lemma 13. The even triangulation I9 [n] with n ≥ 3 can be reduced by using N -flips and 4-contractions. Proof. See the N -flip in the center and the on right-hand side in Fig. 8. Only one N -flip yields an even triangulation that has a twin-contractible part. Now, we have prepared everything for proving Theorem 8. Proof of Theorem 8. By Lemmas 11–13 and [5], every even triangulation on the projective plane can be transformed into one of I1 , I2 , I3 , I4 , I5 , I6 , I7 , I8 , I10 , I11 , I1 + O and I10 by a sequence of N -flips and 4-contractions, where I10 is the one transformed into I1 by a single twin-contraction. According to the proof of Lemma 11, it is clear that I1 + O and I10 are N -equivalent. Moreover, each of them is N -equivalent to I10 . (See the top of Fig. 9.) Next we show that I2 and I4 can be reduced. It is easy to see that the resulting graphs in the figures include such vertices of degree 4. Two remaining ones in Fig. 9 present the facts that I6 ∼ N I8 + O and that the number of vertices of I11 can be reduced. 3. 2-subdivisions of edges Let G be an even triangulation on a closed surface F 2 and let e = ab be an edge of G sharing two triangular faces abc and abd. Put two vertices u, v on the edge ab so that a, u, v, b lie in this order, and join u and v to c and d. We call this operation a 2-subdivision, and {u, v} a 2-subdividing pair. Lemma 14 ([4]). Let K be an even triangulation on any surface F 2 , and let e1 , . . . , e2k be the edges incident to a vertex v of K appearing in the cyclic order around v, where deg(v) = 2k. Let G (resp., G 0 ) be the even triangulation on F 2 obtained from K by a 2-subdivision of ei (resp., ei+2 ). Then G and G 0 are N -equivalent. Lemma 15 ([4]). Let K be an even triangulation on any surface F 2 , and let v be a vertex v of degree 4 with link v1 · · · v4 . Suppose that the link of v1 in K is v2 vv4 x1 · · · x2h−3 , where deg(v1 ) = 2h ≥ 4. Let K 0 be the graph obtained from K by a 4-contraction of v at {v1 , v3 }. Then K can be transformed into the graph obtained from K 0 by adding a 2-subdividing pair to the edge x1 [v1 v3 ], where [v1 v3 ] is the vertex arising from the identification of v1 and v3 . We may suppose that the vertices of a 3-colored even triangulation are colored red, blue and yellow and denote the three color classes by V R (G), VB (G) and VY (G), respectively. We say that an edge with red and blue endpoints is an r b-edge. Similarly, we can define a by-edge and an r y-edge.
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Fig. 9. N -flips in graphs.
Observe that if ei in Lemma 14 is an r b-edge, then so is ei+2 . (Other cases are similar.) Therefore, we can denote an even triangulation obtained from G by adding p, q and r 2-subdividing pairs to an r b-edge, a by-edge and an r y-edge, respectively, by G + r b( p) + by(q) + r y(r ). This notation might express various even triangulations, but they are all N -equivalent, by Lemma 14. Now consider the case when G is non-3-colorable. Recall that a non-3-colorable even triangulation G on the projective plane admits a unique color factor U such that G − U is a non-bipartite even embedding. Let Fr (G) denote the set of edges of G − U , and let Sp(G) = E(G) − Fr (G). If e ∈ Sp(G), we call e an sp-edge, while e ∈ Fr (G) is an f r -edge. Like for the 3-colored case, if ei is an f r -edge (or an sp-edge) in Lemma 14, then so is ei+2 . (In particular, if ei is an sp-edge and so is ei+1 , we can also move the 2-subdividing pair to ei+1 ; otherwise, G − U would be a bipartite even embedding.) Therefore, we can denote an even triangulation obtained from G by adding p and q 2-subdividing pairs to an sp-edge and an f r -edge, respectively, by G + sp( p) + f r (q). It has precisely |U | + p independent vertices and |X | + p + 2q other vertices, and hence has |V (G)| + 2 p + 2q vertices in total. Applying Lemmas 14 and 15, we obtain the following lemma. Lemma 16. Let G be an even triangulation on the projective plane. Suppose that G can be transformed into G 0 by a sequence of 4-contractions and N -flips. (i) If G is 3-colored, then G ∼ N G 0 + r b( p) + by(q) + r y(r ), where |V R (G)| − |V R (G 0 )| = α,
|VB (G)| − |VB (G 0 )| = β,
|VY (G)| − |VY (G 0 )| = γ
and p + r = α, p + q = β, q + r = γ . (ii) If G is not 3-colorable, then G ∼ N G 0 + sp( p) + f r (q), where |U (G)| − |U (G 0 )| = p and |V (G)| − |V (G 0 )| = p + 2q. In Lemma 16, note that if G and G 0 are given, then p, q, r can be determined uniquely.
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Fig. 10. N -flips in I5 + by(1).
4. Proof of Theorems 2 and 4–6 Recall that a 4-contraction reduces the number of vertices by 2, and hence it does not change the parity of the number of vertices. Moreover, an N -flip and a 4-contraction preserve 3-colorability. Therefore, we classify even triangulations on the projective plane into four classes, depending on the 3-colorability and the parity of the number of vertices. By Theorem 8, we have six minimal even triangulations in the projective plane with respect to N -flips and 4contractions. Among them, only I8 is a 3-colorable even triangulation with an even number of vertices. That is, every 3-colorable even triangulation on the projective plane with an even number of vertices can be transformed into I8 by a sequence of N -flips and 4-contractions. Similarly, I3 and I8 are 3-colorable with odd number of vertices. The remaining ones I1 , I7 and I10 are not 3-colorable and have 7, 10 and 10 vertices, respectively. Proof of Theorem 2. Let G be any 3-colored even triangulation on the projective plane. Then, by Theorem 8, there exists a sequence of even triangulations G = G 0 , G 1 , . . . , G k such that G i+1 is obtained from G i by a single N -flip or a single 4-contraction, for i = 0, . . . , k − 1, and G k is one of I3 , I5 and I8 . We first consider the case when |V (G)| is even. Since G k must be isomorphic to I8 , we have G ∼ N I8 + r b( p) + by(q) + r y(r ), where |V R (G)| = p + r + 3, |VB (G)| = p + q + 3 and |VY (G)| = q + r + 4, by Lemma 16. Then this obviously proves the theorem. Now we prove the theorem when |V (G)| is odd. In this case, note that G k is isomorphic to either I3 or I5 . Hence if both G and G 0 can be transformed into exactly one of I3 and I5 by 4-contractions and N -flips, then the same argument as above follows. Let G be a 3-colored even triangulation which is transformed into I5 by 4-contractions and N -flips. Then we have |V R (G)| = p +r + 6, |VB (G)| = p + q + 4 and |VY (G)| = q +r + 3 for some integers p, q, r ≥ 0. It suffices to show that G ∼ N I3 + r b( p 0 ) + by(q 0 ) + r y(r 0 ) for some p 0 , q 0 , r 0 . Observe that I5 + by(1) ∼ N I3 + r b(2) + yr (1), which is shown in Fig. 10. Since G ∼ N I5 + r b( p) + by(q) + r y(r ), we have G ∼ N I3 + r b( p + 2) + by(q − 1) + r y(r + 1), if q ≥ 1. Hence we suppose that q = 0. Then we have |V R (G)| = p + r + 6, |VB (G)| = p + 4 and |VY (G)| = r + 3 for some p, r ≥ 0. Now we prove that there do not exist p 0 , q 0 , r 0 ≥ 0 such that G ∼ N I3 +r b( p 0 ) + by(q 0 ) +r y(r 0 ). For a contradiction, suppose that p 0 , q 0 , r 0 ≥ 0 exist. Since I3 has exactly three yellow vertices, and since |VY (G)| = r + 3, we have q 0 + r 0 = r . Since q 0 , r 0 ≥ 0, we have |q 0 − r 0 | ≤ r . On the other hand, since |V R (G)| − |VB (G)| = r + 2, we have r 0 − q 0 = r + 2, a contradiction. Proof of Theorem 5. It is easy to see that I8 + r b( p) + by(q) + r y(r ) with q ≥ 1 can be transformed into I8 + r b( p + 1) + by(q − 1) + r y(r ) by N - and P2 -flips. Thus, every 3-colored even triangulation on the projective plane with an even number of vertices can be transformed into I8 + r b(m), where |V (G)| = 2m + 10. Similarly, one with an odd number of vertices, except I5 , can be transformed into I3 +r b(m) where |V (G)| = 2m +9. The exception I5 clearly does not have two adjacent vertices of degree 4; thus we can’t apply a P2 -flip to it. Furthermore, we can see that I5 is the face subdivision of K 3,4 . This means that any N -flip to the graph would yield multiple edges. Therefore, we can’t transform I5 into any other graph by the two operations. Then, the theorem holds. Proof of Theorem 4. We use the same method as in the proof of Theorem 2. Let G be a non 3-colorable even triangulation on the projective plane. We first prove the case when |V (G)| is odd. In this case, G is transformed into I1 by a sequence of 4-contractions and N -flips. Therefore, we can prove that G ∼ N I1 + sp( p) + f r (q), where |U (G)| = p + 3 and |X (G)| = p + 2q + 4. When |V (G)| is even, G is transformed into either I7 or I10 by 4-contractions and N -flips, each of which has 10 vertices. If G and G 0 are transformed into one of I7 and I10 , then the same argument as in the proof
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Fig. 11. N -flips in I10 + f r (1).
of Theorem 2 follows. Hence we suppose not. Fig. 11 shows that I10 + f r (1) ∼ N I7 + sp(1). This implies that G ∼ N I10 + sp( p − 1) + f r (q) ∼ N I7 + sp( p) + f r (q − 1), if p, q ≥ 1. Clearly, if one of p = 0 and q = 0 holds, then each of G and G 0 can be transformed into exactly one of I7 and I10 , a contradiction. Proof of Theorem 6. We can prove the theorem by the same method as in the proof of Theorem 5. Consider the minimality of the graphs in Fig. 4; we can’t apply both an N - and a P2 -flip to I7 . References [1] [2] [3] [4] [5]
J.P. Hutchinson, B.R. Richter, P.D. Seymour, Colouring Eulerian triangulations, J. Combin. Theory Ser. B 84 (2002) 225–239. B. Mohar, Coloring Eulerian triangulations of the projective plane, Discrete Math. 244 (2002) 339–343. B. Mohar, C. Thomassen, Graphs on Surfaces, The Johns Hopkins University Press, 2001. A. Nakamoto, T. Sakuma, Y. Suzuki, N -flips in even triangulations on the sphere, J. Graph Theory 51 (2006) 260–268. Y. Suzuki, T. Watanabe, Generating even triangulations of the projective plane, J. Graph Theory 56 (2007) 333–349.
Further reading [1] J.P. Hutchinson, Three-coloring graphs embedded on surfaces with all faces even-sided, J. Combin. Theory Ser. B 65 (1995) 139–155. [2] A. Nakamoto, Quadrangulations on closed surfaces, Interdiscip. Inform. Sci. 7 (2001) 77–98. [3] S. Negami, Diagonal flips of triangulations on surfaces, a survey, Yokohama Math. J. 47 (1999) 1–40.