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Null and non-rainbow colorings of projective plane and sphere triangulations
Discrete Applied Mathematics (
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Null and non-rainbow colorings of projective plane and sphere triangulations Jorge L. Arocha a , Amanda Montejano b,∗ a b
Instituto de Matemáticas, Universidad Nacional Autónoma de México, D.F., Mexico UMDI Facultad de Ciencias, Universidad Nacional Autónoma de México, Querétaro, Mexico
article
info
Article history: Received 14 January 2014 Received in revised form 25 March 2015 Accepted 10 April 2015 Available online xxxx Keywords: Anti-Ramsey theory Non-rainbow colorings Sphere and projective plane triangulations
abstract By considering graphs as topological spaces we introduce, at the level of homology, the notion of a null coloring, which provides new information on the task of clarifying the structure of cycles in a graph. We prove that for any graph G a maximal null coloring f is such that the quotient graph G/f is acyclic. As an application, for maximal planar graphs (sphere triangulations) of order n ≥ 4, we prove that a vertex-coloring containing no rainbow faces uses at most 2n3−1 colors, and this is best possible. For maximal graphs embedded on the projective plane we obtain the analogous best bound 2n3+1 . © 2015 Published by Elsevier B.V.
1. Introduction In this work we consider graphs as topological spaces. A simple graph, which combinatorially is a pair of sets G = (V , E ) (where V is a finite set of elements called vertices, and E is a set of 2-element subsets of V with elements called edges) will be regarded as a subset of points in R3 . In this framework we denote by H1 (G) the first homology group of G, which is a free abelian group with m − n + 1 generators where n and m are respectively the number of vertices and edges of the graph. Roughly speaking, the first homology group H1 (G) of a graph G measures the number of independent 1-dimensional holes (cycles) in the graph. In this work we are interested in connected graphs. In the non connected case we can proceed by considering each component independently. For simple connected graphs we consider vertex colorings, which are not necessarily proper colorings. It is well known that a k-coloring of G can be seen as a homomorphism f : G → Kk , and thus, in our context, as a continuous mapping from G to Kk . Hence, every k-coloring f induces a group homomorphism f∗ : H1 (G) → H1 (Kk ). At this level we introduce a new type of coloring: we say that a vertex k-coloring f of G is a null coloring, if f∗ is zero, that is, all the cycles of G are mapped into closed trivial walks in Kk . Since for any graph G a trivial example of a null coloring is to color all its vertices with one single color, then the interest is on finding null colorings with the maximum number of colors. In Section 2 we give precise definitions. In Section 3 we prove our main theorem, Theorem 1, on maximal null colorings. Finally, in Section 4 we provide as a corollary of Theorem 1, the exact value from which tricolored faces are inevitable in any coloring of a given sphere triangulation, respectively in any coloring of a given projective plane triangulation. The problem of maximizing the number of colors in a vertex coloring avoiding rainbow faces has been studied recently in several papers [3,5,6,8,9]. In Section 4, before stating and proving our result, we summarize some known results in the area.
∗
Corresponding author. E-mail addresses: [email protected] (J.L. Arocha), [email protected] (A. Montejano).
http://dx.doi.org/10.1016/j.dam.2015.04.007 0166-218X/© 2015 Published by Elsevier B.V.
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Fig. 1. A null coloring f such that G/f is not a tree.
2. Null colorings Let G be a simple connected graph. A realization of G is a set of points in a real vector space RN , where each vertex is a different point, and each edge is a straight segment joining its corresponding pair of vertices, where no vertex lies on an edge, except at one of its end-points, and two edges can meet only in a common end-point. It is well known that not every graph can be realized in R2 , but all graphs can be realized in R3 . In this manner we regard a graph G as a topological space: a subset of points in R3 corresponding to a realization of G. Let G be a simple connected graph with |V (G)| = n and |E (G)| = m. Consider a realization G ⊂ R3 and denote by H1 (G) the first homology group of G, which is a free abelian group with m − n + 1 generators. The first homology group, H1 (G), is isomorphic to the group of 1-cycles in G. For instance, if T is a tree, then H1 (T ) ≃ 0, and for any cycle Cn we have H1 (Cn ) ≃ Z. In fact, a basis β of generators for H1 (G) is given by choosing a spanning tree T of G, and an orientation for the edges {e1 , . . . , em−n+1 } of G − T . So, giving a closed walk W in G, it represents the element α1 ⊕· · ·⊕αm−n+1 ∈ Z ⊕· · ·⊕ Z = H1 (G), according with the basis β , where αi is the direct sum of how many times the direct edge ei is transversed by W (for further information on this topic, see [4] and references therein). A k-coloring of G is a surjective mapping f : V (G) → {1, 2, . . . , k}, or equally a partition of V (G) into exactly k nonempty parts called color classes. It is well known that a k-coloring of a graph G can be viewed as a homomorphism f : G → Kk (note that since we allow colorings which are not proper the homomorphism may be reflexive). By considering graphs as topological spaces we can think on a k-coloring of G as a continuous mapping f : G → Kk by sending each vertex of G to its image in Kk and extending the map linearly to the edges. Thus, at the level of homology, a k-coloring f induces a group homomorphism f∗ : H1 (G) → H1 (Kk ). Such a group homomorphism is called zero, if for every c ∈ H1 (G), then f∗ (c ) = 0 ∈ H1 (Kk ). Definition 1. For a simple connected graph G, a k-coloring f : G → Kk is called a null coloring, if and only if, f∗ : H1 (G) → H1 (Kk ) is zero. Let f be a coloring of G, we define the quotient graph G/f as the graph with vertices being the color classes, and two color classes are adjacent, if there is an edge in G which incident vertices have those colors. Naturally, if a coloring f of a connected graph G satisfies that G/f is a tree, then f is null. The opposite is not true as shown in the example illustrated in Fig. 1 (the coloring depicted is null since every cycle in G is mapped into a close trivial walk in G/f , that is, a walk that topologically can be reduced to a point in G/f ). 3. Maximal null colorings Let G be a connected graph and f be a k-coloring of G. We called f a maximal null coloring if f is a null coloring and there are no null (k + 1)-colorings of G. As shown in Fig. 1, not every null coloring f satisfies that the quotient graph G/f is acyclic. Nevertheless, this is true for maximal null colorings. That is, a maximal null coloring satisfies that its quotient graph is acyclic as we will prove next. Theorem 1. Let G be a simple connected graph, and f be a maximal null coloring of G. Then G/f is a tree. We first prove some lemmas. Let G be a simple connected graph and f be a coloring of G. Let u, v ∈ V (G) be two vertices with the same color, that is f (u) = f (v). Denote by G′ the graph which is obtained from G by identifying the vertices u and v , and denote this vertex of G′ by uv . Let h : G → G′ be the corresponding homomorphism. The coloring f of G naturally induces a coloring f ′ of G′ by defining f ′ (uv) = f (u) = f (v) and f ′ (x) = f (x) if x ̸∈ {u, v}. Obviously G/f = G′ /f ′ . We denote by d(u, v) the distance between u and v in G. Lemma 1. For a coloring f of G, and two vertices u, v ∈ V (G) with the same color, let G′ , f ′ , and h : G → G′ as defined above. If d(u, v) ≤ 2, then h∗ : H1 (G) → H1 (G′ ) is an epimorphism.
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Proof. Let γ be a cycle of G′ . We consider the following cases: (1) γ does not contain uv . Then, obviously the corresponding cycle of G is sent, by h, to γ . (2) γ = {uv, z1 , z2 , . . . , zρ , uv} and either {u, z1 , z2 , . . . , zρ , u} or {v, z1 , z2 , . . . , zρ , v} is a cycle in G. Again the corresponding cycle is sent, by h, to γ . (3) γ = {uv, z1 , z2 , . . . , zρ , uv} and {u, z1 , z2 , . . . , zρ , v} is a path in G. If u and v are adjacent vertices in G, then at the level of homology, the element of H1 (G) induced by the cycle {u, z1 , z2 , . . . , zρ , v, u} is sent, by h∗ , to the element of H1 (G′ ) induced by γ . If d(u, v) = 2 in G, let w ∈ V (G) be a vertex adjacent to both u and v . Suppose first that w ̸= zi for any 1 ≤ i ≤ ρ , then consider the cycle γ 0 of G, γ 0 = {u, z1 , z2 , . . . , zρ , v, w, u}. Clearly, at the level of homology, the element of H1 (G) induced by γ 0 is sent, by h∗ , to the element of H1 (G′ ) induced by γ . Suppose now that w = zi0 for some 1 ≤ i0 ≤ ρ . Let γ 1 = {u, z1 , z2 , . . . , zi0 = w, u} and γ 2 = {v, w = zi0 , zi0 +1 , . . . , zρ , v}. Note that, at the level of homology, the sum of the elements of H1 (G) induced by γ 1 and γ 2 is sent, by h∗ , to the element of H1 (G′ ) induced by γ . In any case, h∗ : H1 (G) → H1 (G′ ) is an epimorphism. Lemma 2. For a maximal null coloring f of G, and two vertices u, v ∈ V (G) with the same color, let G′ , f ′ , and h : G → G′ as defined above. If d(u, v) ≤ 2, then f ′ is a maximal null coloring of G′ . Proof. By Lemma 1, f ′ is a null coloring of G′ . We shall prove that f ′ is a maximal null coloring. Suppose there exists a null coloring g ′ of G′ using more colors than f ′ . Then we extend g ′ to a coloring g of G, by letting g (u) = g (v) = g ′ (uv). Since, at the level of homology, g∗ = g∗′ ◦ h∗ , then g is a null coloring of G, contradicting the maximality of f . Given a coloring f of a simple connected graph G and two different colors i, j, we called an edge e ∈ E (G) an (ij)-edge if the vertices of e have colors i and j respectively. Lemma 3. Let f : G → Kk be a null coloring. For any pair of different colors i, j ∈ V (Kk ) and any cycle C of G, the number of (ij)-edges in C is even. Proof. Let T be a spanning tree of Kk such that the edge (i, j) ̸∈ E (T ), and let β be a basis of H1 (Kk ) given by some orientation of E (Kk ) \ E (T ). Since f is a null coloring, then the closed path f (C ) must represent the element 0 ⊕ · · · ⊕ 0 ∈ Z ⊕ · · · ⊕ Z = H1 (Kk ). Hence the number of times that the path f (C ) crosses the edge (i, j) ∈ E (Kk ) is even, as any time that (i, j) is taken in one direction it has to be taken in the other direction at some point in the path. Lemma 4. Let f be a maximal null coloring of G. If any two vertices u, v ∈ V (G) with the same color satisfy d(u, v) ≥ 2, then G is a bipartite graph. Proof. By assumption there are no monochromatic edges, and by Lemma 3 the number of dichromatic edges in any cycle is even. Thus, any cycle of G has an even number of vertices. Now, we are ready to prove Theorem 1. Proof of Theorem 1. The proof is by induction on the order of G. The theorem is straightforward for graphs with less that four vertices. Suppose there is a minimum n such that there exists a simple connected graph G of order n contradicting the theorem. Assume first that there are two vertices u, v ∈ V (G) with f (u) = f (v) and d(u, v) ≤ 2. By Lemma 2, f ′ is a maximal null coloring of G′ . Since G/f = G′ /f ′ and G′ has n − 1 vertices, we get a contradiction. Assume now that for any pair of vertices u, v ∈ V (G) with f (u) = f (v) then d(u, v) > 2. Then G is bipartite according to Lemma 4. So, there is an independent set I ⊂ V (G) such that |I | ≥ ⌈ 2n ⌉, where |V (G)| = n, from which we conclude that f uses at least ⌈ 2n ⌉ + 1 colors, since we can define a null coloring of G by assigning to each vertex of I a distinct color, and to all
vertices in V (G) \ I one more color. Therefore, by the pigeonhole principle there must be a vertex w ∈ V (G) such that f (w) is unique. Next, we shall prove that w is either a cut point or a terminal point of G. Suppose there is a cycle γ of G through w, γ = {w, z1 , z2 , . . . , zρ , w}. Since f (w) is unique, Lemma 3 implies that f (z1 ) = f (zρ ), a contradiction as d(z1 , zρ ) = 2. Now, let deg (w) = r be the number of neighbors of w in G, and let G1 , G2 , . . . ,Gr be the connected components of G − w . Let f1 , . . . ,fr be the restrictions of f to each G1 , . . . ,Gr . It is not difficult to see that every fi is a maximal null coloring. So, by induction Gi /fi is a tree for every 1 ≤ i ≤ r. Note also that each fi uses distinct colors (otherwise f would not be maximal). Then we conclude that G/f is a tree. 4. Non-rainbow colorings As an application of Theorem 1, in this section we give sharp bounds on the maximum number of colors in a non-rainbow coloring (see definition below) of a sphere triangulation or a projective plane triangulation. Giving a connected planar graph G and a vertex-coloring f of G we say that a face of G is rainbow (with respect to f ) if its vertices receive pairwise distinct colors. A coloring which contains no rainbow faces is called a non-rainbow coloring. The
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Fig. 2. A non-rainbow coloring which is not a null coloring.
largest integer for which there is a non-rainbow coloring of G is denoted by χf (G). Before summarizing some known results on the study of this parameter, we shall observe that there exists a non-rainbow k-coloring of G for every k ≤ χf (G), while all k-colorings of G with k > χf (G) contain a rainbow face. Thus, χf (G) + 1 is the minimum integer for which every coloring of G contains a rainbow face. In this sense the problem is considered an extremal anti-Ramsey problem. Ramamurthi and West [9] provided a natural lower bond for χf (G) in terms of the independent number of G. By the four color theorem it follows from that natural bound that: χf (G) ≥ ⌈ nk ⌉ + 1, where k = 2, 3 or 4, depending on whether G is a 2-, 3-, or 4-chromatic graph. In the same paper it was proved that the above lower bound is tight for k = 2, or k = 3, and it is within one of being tight for k = 4. In [7] the authors showed tightness in this latter case. Jendrol’ [5] studied bounds of χf (G) for cubic plane graphs in terms of the independence, and the edge-independence number of the dual graph G∗ . Jungić, Král’and Skrekovski [6] investigated the problem for triangle-free planar graphs and provided sharp lower bounds in terms of the girth g, for every g ≥ 4. Dvořák, Král’ and Škrekovski [3] studied upper bounds for 3-, 4- and 5-connected planar graphs. Particularly for a 3-connected planar graph G they obtain the best possible upper bound
χf (G) ≤
7n − 8 9
.
(1)
Related works concerning graphs embedding in other surfaces are [1,2,8]. In this work we investigate the problem of determining a sharp upper bound for χf (G) concerning maximal planar graphs also called sphere triangulations, which is a special subclass of 3-connected planar graphs. By means of Theorem 1 we provide a sharp upper bound slightly smaller that the one in (1). Theorem 2. A non-rainbow coloring of a maximal planar graph with n ≥ 4 vertices uses at most possible.
2n−1 3
colors, and this is best
By using the same techniques, for maximal graphs embedded in the projective plane also called projective plane triangulations, we obtain the analogous sharp upper bound. Theorem 3. A non-rainbow coloring of a maximal graph embedded on a projective plane with n ≥ 4 vertices uses at most colors, and this is best possible.
2n+1 3
We deduce both results from Theorem 1 by proving that, for maximal planar graphs, or maximal graphs embedded in the projective plane, a maximal non-rainbow coloring is a maximal null coloring. First of all, note that a null coloring f is necessarily a non-rainbow coloring, but the opposite in general is not true as shown in the example illustrated in Fig. 2. Nevertheless, this is true for maximal planar graphs. Lemma 5. Let T be a maximal graph embedded on a sphere or a projective plane. Then a non-rainbow coloring of T is also a null coloring of T . Proof. Let S denote either a sphere or a projective plane. Note that a maximal graph T embedded on S is the 1-skeleton of a triangulation of S. Let f be a non-rainbow k-coloring of G. Recall that f may be thought of as a continuous mapping from T to Kk . The fact that f is a non-rainbow coloring allows us to extend the map f , triangle by triangle, to a map F : S → Kk . We shall note that, the level of homology, F∗ : H1 (S ) → H1 (Kk ) is zero, since H1 (S ) is either zero or Z2 and H1 (Kk ) is free abelian. Furthermore, if we denote by i : T → S the inclusion, then F ◦ i = f . So, by functoriality F∗ ◦ i∗ = f∗ , and therefore f∗ : H1 (T ) → H1 (Kk ) is zero. Thus, f is a null coloring of T as claimed. Note that the converse of the previous lemma is obviously true, from which it follows directly the next lemma. Lemma 6. Let T be a maximal graph embedded on a sphere, or a projective plane. Then a maximal non-rainbow coloring of T is also a maximal null coloring of T .
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Now, by means of Theorem 1 and Lemma 6 we can prove Theorems 2 and 3. Proof of Theorem 2. Let S be a sphere, and let T be a maximal planar graph embedded on S with |V (T )| = n ≥ 4. Let χf (T ) = k, and consider f a non-rainbow k-coloring of T . By definition f is a maximal non-rainbow coloring of T , and thus, by Lemma 6, f is a maximal null coloring of T . Then, Theorem 1 implies that T /f is a tree. Note that T /f has k vertices, and so it has k − 1 edges. It is not difficult to see that, for each pair of adjacent color classes there are at least three bi-chromatic faces using those colors. Thus, the total number of faces inT , which is 2n − 4 according to Euler’s formula, must be greater than 3(k − 1). From this follows that k = χf (T ) ≤ 2n3−1 , which concludes the proof of the theorem for maximal planar graphs.
Proof of Theorem 3. If S is a projective plane, the proof is completely analogous except that the number of faces in a maximal graph of order n ≥ 4 embedded on S, is 2n − 2, which leads to the upper bound k = χf (T ) ≤ 2n3+1 . To prove the sharpness of Theorems 2 and 3 we need to exhibit, for any n ≥ 4, a maximal planar graph T (respectively a maximal graph embedded in the projective plane) with n vertices, and a non-rainbow coloring of T using 2n3−1
(respectively 2n3+1 ) colors. This can be done by taking face subdivisions of suitable given sphere or projective plane triangulations. Let S denote a surface which is either a sphere or a projective plane, and let T be a maximal planar graph embedded on S. To subdivide a face {v1 , v2 , v3 } of T is to add a new vertex u and the three edges uv1 , uv2 and uv3 . This operation results in a new triangulation of S.
Proposition 1. For every n ≥ 4, there is a sphere triangulation with n vertices that has a non-rainbow coloring with ⌊ 2n3−1 ⌋ colors. Proof. Let Tk be a sphere triangulation with k = n − ⌊ 2n3−1 ⌋ + 1 vertices. Let T be the sphere triangulation obtained by subdividing ⌊ 2n3−1 ⌋ − 1 faces of Tk .
Note that T has exactly n vertices. Now, let us give a non-rainbow coloring of T with ⌊ 2n3−1 ⌋ colors. Let the original vertices of Tk be colored by one single color, and each new vertex be colored with a different color. Proposition 2. For every n ≥ 4, there is a projective plane triangulation with n vertices that has a non-rainbow coloring with ⌊ 2n3+1 ⌋ colors.
Proof. Just start with a projective plane triangulation with k = n − ⌊ 2n3+1 ⌋ + 1 vertices.
Acknowledgments J.L. Arocha was partially supported by CONACyT: 166951, and A. Montejano was partially supported by CONACyT: 166306 and 219827; PAPIIT: IA102013. We also acknowledge support for Center of Innovation in Mathematics, CINNMA. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
J.L. Arocha, J. Bracho, V. Neumann-Lara, On the minimum size of tight hypergraphs, J. Graph Theory 16 (1992) 319–326. J.L. Arocha, J. Bracho, V. Neumann-Lara, Tight and untight triangulations of surfaces by complete graphs, J. Combin. Theory Ser. B 63 (1995) 185–199. Z. Devořák, D. Král’, R. Škrekovski, Non-rainbow colorings of 3-, 4- and 5-connected plane graphs, J. Graph Theory (2009) 130–145. P. Gibling, Graphs, Surfaces and Homology, third ed., Cambridge University Press, 2010. S. Jendrol’, Rainbowness of cubic plane graphs, Discrete Math 306 (2006) 3321–3326. V. Jungić, D. Král’, R. Skrekovski, Coloring of plane graphs with no rainbow faces, Combinatorica 26 (2) (2006) 169–182. A. Montejano, D. Ventura, A note on non-rainbow colorings concerning 4-chromatic planar graphs (in preparation). Seiya Negami, Looseness ranges of triangulations on close surfaces, Discrete Math. 303 (2005) 167–174. R. Ramamurthi, D.B. West, Maximum face-constrained coloring of plane graphs, Discrete Math. 274 (13) (2004) 233–240.
)
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Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Null and non-rainbow colorings of projective plane and sphere triangulations Jorge L. Arocha a , Amanda Montejano b,∗ a b
Instituto de Matemáticas, Universidad Nacional Autónoma de México, D.F., Mexico UMDI Facultad de Ciencias, Universidad Nacional Autónoma de México, Querétaro, Mexico
article
info
Article history: Received 14 January 2014 Received in revised form 25 March 2015 Accepted 10 April 2015 Available online xxxx Keywords: Anti-Ramsey theory Non-rainbow colorings Sphere and projective plane triangulations
abstract By considering graphs as topological spaces we introduce, at the level of homology, the notion of a null coloring, which provides new information on the task of clarifying the structure of cycles in a graph. We prove that for any graph G a maximal null coloring f is such that the quotient graph G/f is acyclic. As an application, for maximal planar graphs (sphere triangulations) of order n ≥ 4, we prove that a vertex-coloring containing no rainbow faces uses at most 2n3−1 colors, and this is best possible. For maximal graphs embedded on the projective plane we obtain the analogous best bound 2n3+1 . © 2015 Published by Elsevier B.V.
1. Introduction In this work we consider graphs as topological spaces. A simple graph, which combinatorially is a pair of sets G = (V , E ) (where V is a finite set of elements called vertices, and E is a set of 2-element subsets of V with elements called edges) will be regarded as a subset of points in R3 . In this framework we denote by H1 (G) the first homology group of G, which is a free abelian group with m − n + 1 generators where n and m are respectively the number of vertices and edges of the graph. Roughly speaking, the first homology group H1 (G) of a graph G measures the number of independent 1-dimensional holes (cycles) in the graph. In this work we are interested in connected graphs. In the non connected case we can proceed by considering each component independently. For simple connected graphs we consider vertex colorings, which are not necessarily proper colorings. It is well known that a k-coloring of G can be seen as a homomorphism f : G → Kk , and thus, in our context, as a continuous mapping from G to Kk . Hence, every k-coloring f induces a group homomorphism f∗ : H1 (G) → H1 (Kk ). At this level we introduce a new type of coloring: we say that a vertex k-coloring f of G is a null coloring, if f∗ is zero, that is, all the cycles of G are mapped into closed trivial walks in Kk . Since for any graph G a trivial example of a null coloring is to color all its vertices with one single color, then the interest is on finding null colorings with the maximum number of colors. In Section 2 we give precise definitions. In Section 3 we prove our main theorem, Theorem 1, on maximal null colorings. Finally, in Section 4 we provide as a corollary of Theorem 1, the exact value from which tricolored faces are inevitable in any coloring of a given sphere triangulation, respectively in any coloring of a given projective plane triangulation. The problem of maximizing the number of colors in a vertex coloring avoiding rainbow faces has been studied recently in several papers [3,5,6,8,9]. In Section 4, before stating and proving our result, we summarize some known results in the area.
∗
Corresponding author. E-mail addresses: [email protected] (J.L. Arocha), [email protected] (A. Montejano).
http://dx.doi.org/10.1016/j.dam.2015.04.007 0166-218X/© 2015 Published by Elsevier B.V.
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Fig. 1. A null coloring f such that G/f is not a tree.
2. Null colorings Let G be a simple connected graph. A realization of G is a set of points in a real vector space RN , where each vertex is a different point, and each edge is a straight segment joining its corresponding pair of vertices, where no vertex lies on an edge, except at one of its end-points, and two edges can meet only in a common end-point. It is well known that not every graph can be realized in R2 , but all graphs can be realized in R3 . In this manner we regard a graph G as a topological space: a subset of points in R3 corresponding to a realization of G. Let G be a simple connected graph with |V (G)| = n and |E (G)| = m. Consider a realization G ⊂ R3 and denote by H1 (G) the first homology group of G, which is a free abelian group with m − n + 1 generators. The first homology group, H1 (G), is isomorphic to the group of 1-cycles in G. For instance, if T is a tree, then H1 (T ) ≃ 0, and for any cycle Cn we have H1 (Cn ) ≃ Z. In fact, a basis β of generators for H1 (G) is given by choosing a spanning tree T of G, and an orientation for the edges {e1 , . . . , em−n+1 } of G − T . So, giving a closed walk W in G, it represents the element α1 ⊕· · ·⊕αm−n+1 ∈ Z ⊕· · ·⊕ Z = H1 (G), according with the basis β , where αi is the direct sum of how many times the direct edge ei is transversed by W (for further information on this topic, see [4] and references therein). A k-coloring of G is a surjective mapping f : V (G) → {1, 2, . . . , k}, or equally a partition of V (G) into exactly k nonempty parts called color classes. It is well known that a k-coloring of a graph G can be viewed as a homomorphism f : G → Kk (note that since we allow colorings which are not proper the homomorphism may be reflexive). By considering graphs as topological spaces we can think on a k-coloring of G as a continuous mapping f : G → Kk by sending each vertex of G to its image in Kk and extending the map linearly to the edges. Thus, at the level of homology, a k-coloring f induces a group homomorphism f∗ : H1 (G) → H1 (Kk ). Such a group homomorphism is called zero, if for every c ∈ H1 (G), then f∗ (c ) = 0 ∈ H1 (Kk ). Definition 1. For a simple connected graph G, a k-coloring f : G → Kk is called a null coloring, if and only if, f∗ : H1 (G) → H1 (Kk ) is zero. Let f be a coloring of G, we define the quotient graph G/f as the graph with vertices being the color classes, and two color classes are adjacent, if there is an edge in G which incident vertices have those colors. Naturally, if a coloring f of a connected graph G satisfies that G/f is a tree, then f is null. The opposite is not true as shown in the example illustrated in Fig. 1 (the coloring depicted is null since every cycle in G is mapped into a close trivial walk in G/f , that is, a walk that topologically can be reduced to a point in G/f ). 3. Maximal null colorings Let G be a connected graph and f be a k-coloring of G. We called f a maximal null coloring if f is a null coloring and there are no null (k + 1)-colorings of G. As shown in Fig. 1, not every null coloring f satisfies that the quotient graph G/f is acyclic. Nevertheless, this is true for maximal null colorings. That is, a maximal null coloring satisfies that its quotient graph is acyclic as we will prove next. Theorem 1. Let G be a simple connected graph, and f be a maximal null coloring of G. Then G/f is a tree. We first prove some lemmas. Let G be a simple connected graph and f be a coloring of G. Let u, v ∈ V (G) be two vertices with the same color, that is f (u) = f (v). Denote by G′ the graph which is obtained from G by identifying the vertices u and v , and denote this vertex of G′ by uv . Let h : G → G′ be the corresponding homomorphism. The coloring f of G naturally induces a coloring f ′ of G′ by defining f ′ (uv) = f (u) = f (v) and f ′ (x) = f (x) if x ̸∈ {u, v}. Obviously G/f = G′ /f ′ . We denote by d(u, v) the distance between u and v in G. Lemma 1. For a coloring f of G, and two vertices u, v ∈ V (G) with the same color, let G′ , f ′ , and h : G → G′ as defined above. If d(u, v) ≤ 2, then h∗ : H1 (G) → H1 (G′ ) is an epimorphism.
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Proof. Let γ be a cycle of G′ . We consider the following cases: (1) γ does not contain uv . Then, obviously the corresponding cycle of G is sent, by h, to γ . (2) γ = {uv, z1 , z2 , . . . , zρ , uv} and either {u, z1 , z2 , . . . , zρ , u} or {v, z1 , z2 , . . . , zρ , v} is a cycle in G. Again the corresponding cycle is sent, by h, to γ . (3) γ = {uv, z1 , z2 , . . . , zρ , uv} and {u, z1 , z2 , . . . , zρ , v} is a path in G. If u and v are adjacent vertices in G, then at the level of homology, the element of H1 (G) induced by the cycle {u, z1 , z2 , . . . , zρ , v, u} is sent, by h∗ , to the element of H1 (G′ ) induced by γ . If d(u, v) = 2 in G, let w ∈ V (G) be a vertex adjacent to both u and v . Suppose first that w ̸= zi for any 1 ≤ i ≤ ρ , then consider the cycle γ 0 of G, γ 0 = {u, z1 , z2 , . . . , zρ , v, w, u}. Clearly, at the level of homology, the element of H1 (G) induced by γ 0 is sent, by h∗ , to the element of H1 (G′ ) induced by γ . Suppose now that w = zi0 for some 1 ≤ i0 ≤ ρ . Let γ 1 = {u, z1 , z2 , . . . , zi0 = w, u} and γ 2 = {v, w = zi0 , zi0 +1 , . . . , zρ , v}. Note that, at the level of homology, the sum of the elements of H1 (G) induced by γ 1 and γ 2 is sent, by h∗ , to the element of H1 (G′ ) induced by γ . In any case, h∗ : H1 (G) → H1 (G′ ) is an epimorphism. Lemma 2. For a maximal null coloring f of G, and two vertices u, v ∈ V (G) with the same color, let G′ , f ′ , and h : G → G′ as defined above. If d(u, v) ≤ 2, then f ′ is a maximal null coloring of G′ . Proof. By Lemma 1, f ′ is a null coloring of G′ . We shall prove that f ′ is a maximal null coloring. Suppose there exists a null coloring g ′ of G′ using more colors than f ′ . Then we extend g ′ to a coloring g of G, by letting g (u) = g (v) = g ′ (uv). Since, at the level of homology, g∗ = g∗′ ◦ h∗ , then g is a null coloring of G, contradicting the maximality of f . Given a coloring f of a simple connected graph G and two different colors i, j, we called an edge e ∈ E (G) an (ij)-edge if the vertices of e have colors i and j respectively. Lemma 3. Let f : G → Kk be a null coloring. For any pair of different colors i, j ∈ V (Kk ) and any cycle C of G, the number of (ij)-edges in C is even. Proof. Let T be a spanning tree of Kk such that the edge (i, j) ̸∈ E (T ), and let β be a basis of H1 (Kk ) given by some orientation of E (Kk ) \ E (T ). Since f is a null coloring, then the closed path f (C ) must represent the element 0 ⊕ · · · ⊕ 0 ∈ Z ⊕ · · · ⊕ Z = H1 (Kk ). Hence the number of times that the path f (C ) crosses the edge (i, j) ∈ E (Kk ) is even, as any time that (i, j) is taken in one direction it has to be taken in the other direction at some point in the path. Lemma 4. Let f be a maximal null coloring of G. If any two vertices u, v ∈ V (G) with the same color satisfy d(u, v) ≥ 2, then G is a bipartite graph. Proof. By assumption there are no monochromatic edges, and by Lemma 3 the number of dichromatic edges in any cycle is even. Thus, any cycle of G has an even number of vertices. Now, we are ready to prove Theorem 1. Proof of Theorem 1. The proof is by induction on the order of G. The theorem is straightforward for graphs with less that four vertices. Suppose there is a minimum n such that there exists a simple connected graph G of order n contradicting the theorem. Assume first that there are two vertices u, v ∈ V (G) with f (u) = f (v) and d(u, v) ≤ 2. By Lemma 2, f ′ is a maximal null coloring of G′ . Since G/f = G′ /f ′ and G′ has n − 1 vertices, we get a contradiction. Assume now that for any pair of vertices u, v ∈ V (G) with f (u) = f (v) then d(u, v) > 2. Then G is bipartite according to Lemma 4. So, there is an independent set I ⊂ V (G) such that |I | ≥ ⌈ 2n ⌉, where |V (G)| = n, from which we conclude that f uses at least ⌈ 2n ⌉ + 1 colors, since we can define a null coloring of G by assigning to each vertex of I a distinct color, and to all
vertices in V (G) \ I one more color. Therefore, by the pigeonhole principle there must be a vertex w ∈ V (G) such that f (w) is unique. Next, we shall prove that w is either a cut point or a terminal point of G. Suppose there is a cycle γ of G through w, γ = {w, z1 , z2 , . . . , zρ , w}. Since f (w) is unique, Lemma 3 implies that f (z1 ) = f (zρ ), a contradiction as d(z1 , zρ ) = 2. Now, let deg (w) = r be the number of neighbors of w in G, and let G1 , G2 , . . . ,Gr be the connected components of G − w . Let f1 , . . . ,fr be the restrictions of f to each G1 , . . . ,Gr . It is not difficult to see that every fi is a maximal null coloring. So, by induction Gi /fi is a tree for every 1 ≤ i ≤ r. Note also that each fi uses distinct colors (otherwise f would not be maximal). Then we conclude that G/f is a tree. 4. Non-rainbow colorings As an application of Theorem 1, in this section we give sharp bounds on the maximum number of colors in a non-rainbow coloring (see definition below) of a sphere triangulation or a projective plane triangulation. Giving a connected planar graph G and a vertex-coloring f of G we say that a face of G is rainbow (with respect to f ) if its vertices receive pairwise distinct colors. A coloring which contains no rainbow faces is called a non-rainbow coloring. The
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Fig. 2. A non-rainbow coloring which is not a null coloring.
largest integer for which there is a non-rainbow coloring of G is denoted by χf (G). Before summarizing some known results on the study of this parameter, we shall observe that there exists a non-rainbow k-coloring of G for every k ≤ χf (G), while all k-colorings of G with k > χf (G) contain a rainbow face. Thus, χf (G) + 1 is the minimum integer for which every coloring of G contains a rainbow face. In this sense the problem is considered an extremal anti-Ramsey problem. Ramamurthi and West [9] provided a natural lower bond for χf (G) in terms of the independent number of G. By the four color theorem it follows from that natural bound that: χf (G) ≥ ⌈ nk ⌉ + 1, where k = 2, 3 or 4, depending on whether G is a 2-, 3-, or 4-chromatic graph. In the same paper it was proved that the above lower bound is tight for k = 2, or k = 3, and it is within one of being tight for k = 4. In [7] the authors showed tightness in this latter case. Jendrol’ [5] studied bounds of χf (G) for cubic plane graphs in terms of the independence, and the edge-independence number of the dual graph G∗ . Jungić, Král’and Skrekovski [6] investigated the problem for triangle-free planar graphs and provided sharp lower bounds in terms of the girth g, for every g ≥ 4. Dvořák, Král’ and Škrekovski [3] studied upper bounds for 3-, 4- and 5-connected planar graphs. Particularly for a 3-connected planar graph G they obtain the best possible upper bound
χf (G) ≤
7n − 8 9
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(1)
Related works concerning graphs embedding in other surfaces are [1,2,8]. In this work we investigate the problem of determining a sharp upper bound for χf (G) concerning maximal planar graphs also called sphere triangulations, which is a special subclass of 3-connected planar graphs. By means of Theorem 1 we provide a sharp upper bound slightly smaller that the one in (1). Theorem 2. A non-rainbow coloring of a maximal planar graph with n ≥ 4 vertices uses at most possible.
2n−1 3
colors, and this is best
By using the same techniques, for maximal graphs embedded in the projective plane also called projective plane triangulations, we obtain the analogous sharp upper bound. Theorem 3. A non-rainbow coloring of a maximal graph embedded on a projective plane with n ≥ 4 vertices uses at most colors, and this is best possible.
2n+1 3
We deduce both results from Theorem 1 by proving that, for maximal planar graphs, or maximal graphs embedded in the projective plane, a maximal non-rainbow coloring is a maximal null coloring. First of all, note that a null coloring f is necessarily a non-rainbow coloring, but the opposite in general is not true as shown in the example illustrated in Fig. 2. Nevertheless, this is true for maximal planar graphs. Lemma 5. Let T be a maximal graph embedded on a sphere or a projective plane. Then a non-rainbow coloring of T is also a null coloring of T . Proof. Let S denote either a sphere or a projective plane. Note that a maximal graph T embedded on S is the 1-skeleton of a triangulation of S. Let f be a non-rainbow k-coloring of G. Recall that f may be thought of as a continuous mapping from T to Kk . The fact that f is a non-rainbow coloring allows us to extend the map f , triangle by triangle, to a map F : S → Kk . We shall note that, the level of homology, F∗ : H1 (S ) → H1 (Kk ) is zero, since H1 (S ) is either zero or Z2 and H1 (Kk ) is free abelian. Furthermore, if we denote by i : T → S the inclusion, then F ◦ i = f . So, by functoriality F∗ ◦ i∗ = f∗ , and therefore f∗ : H1 (T ) → H1 (Kk ) is zero. Thus, f is a null coloring of T as claimed. Note that the converse of the previous lemma is obviously true, from which it follows directly the next lemma. Lemma 6. Let T be a maximal graph embedded on a sphere, or a projective plane. Then a maximal non-rainbow coloring of T is also a maximal null coloring of T .
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Now, by means of Theorem 1 and Lemma 6 we can prove Theorems 2 and 3. Proof of Theorem 2. Let S be a sphere, and let T be a maximal planar graph embedded on S with |V (T )| = n ≥ 4. Let χf (T ) = k, and consider f a non-rainbow k-coloring of T . By definition f is a maximal non-rainbow coloring of T , and thus, by Lemma 6, f is a maximal null coloring of T . Then, Theorem 1 implies that T /f is a tree. Note that T /f has k vertices, and so it has k − 1 edges. It is not difficult to see that, for each pair of adjacent color classes there are at least three bi-chromatic faces using those colors. Thus, the total number of faces inT , which is 2n − 4 according to Euler’s formula, must be greater than 3(k − 1). From this follows that k = χf (T ) ≤ 2n3−1 , which concludes the proof of the theorem for maximal planar graphs.
Proof of Theorem 3. If S is a projective plane, the proof is completely analogous except that the number of faces in a maximal graph of order n ≥ 4 embedded on S, is 2n − 2, which leads to the upper bound k = χf (T ) ≤ 2n3+1 . To prove the sharpness of Theorems 2 and 3 we need to exhibit, for any n ≥ 4, a maximal planar graph T (respectively a maximal graph embedded in the projective plane) with n vertices, and a non-rainbow coloring of T using 2n3−1
(respectively 2n3+1 ) colors. This can be done by taking face subdivisions of suitable given sphere or projective plane triangulations. Let S denote a surface which is either a sphere or a projective plane, and let T be a maximal planar graph embedded on S. To subdivide a face {v1 , v2 , v3 } of T is to add a new vertex u and the three edges uv1 , uv2 and uv3 . This operation results in a new triangulation of S.
Proposition 1. For every n ≥ 4, there is a sphere triangulation with n vertices that has a non-rainbow coloring with ⌊ 2n3−1 ⌋ colors. Proof. Let Tk be a sphere triangulation with k = n − ⌊ 2n3−1 ⌋ + 1 vertices. Let T be the sphere triangulation obtained by subdividing ⌊ 2n3−1 ⌋ − 1 faces of Tk .
Note that T has exactly n vertices. Now, let us give a non-rainbow coloring of T with ⌊ 2n3−1 ⌋ colors. Let the original vertices of Tk be colored by one single color, and each new vertex be colored with a different color. Proposition 2. For every n ≥ 4, there is a projective plane triangulation with n vertices that has a non-rainbow coloring with ⌊ 2n3+1 ⌋ colors.
Proof. Just start with a projective plane triangulation with k = n − ⌊ 2n3+1 ⌋ + 1 vertices.
Acknowledgments J.L. Arocha was partially supported by CONACyT: 166951, and A. Montejano was partially supported by CONACyT: 166306 and 219827; PAPIIT: IA102013. We also acknowledge support for Center of Innovation in Mathematics, CINNMA. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
J.L. Arocha, J. Bracho, V. Neumann-Lara, On the minimum size of tight hypergraphs, J. Graph Theory 16 (1992) 319–326. J.L. Arocha, J. Bracho, V. Neumann-Lara, Tight and untight triangulations of surfaces by complete graphs, J. Combin. Theory Ser. B 63 (1995) 185–199. Z. Devořák, D. Král’, R. Škrekovski, Non-rainbow colorings of 3-, 4- and 5-connected plane graphs, J. Graph Theory (2009) 130–145. P. Gibling, Graphs, Surfaces and Homology, third ed., Cambridge University Press, 2010. S. Jendrol’, Rainbowness of cubic plane graphs, Discrete Math 306 (2006) 3321–3326. V. Jungić, D. Král’, R. Skrekovski, Coloring of plane graphs with no rainbow faces, Combinatorica 26 (2) (2006) 169–182. A. Montejano, D. Ventura, A note on non-rainbow colorings concerning 4-chromatic planar graphs (in preparation). Seiya Negami, Looseness ranges of triangulations on close surfaces, Discrete Math. 303 (2005) 167–174. R. Ramamurthi, D.B. West, Maximum face-constrained coloring of plane graphs, Discrete Math. 274 (13) (2004) 233–240.