- Email: [email protected]
Facial edge ranking of plane graphs
Discrete Applied Mathematics 194 (2015) 60–64
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Facial edge ranking of plane graphs Július Czap a,∗ , Stanislav Jendrol’ b a
Department of Applied Mathematics and Business Informatics, Faculty of Economics, Technical University of Košice, Němcovej 32, 040 01 Košice, Slovakia b Institute of Mathematics, P. J. Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
article
info
Article history: Received 22 May 2014 Received in revised form 13 March 2015 Accepted 5 May 2015 Available online 27 May 2015 Keywords: Facial edge ranking Plane graph
abstract A facial edge k-ranking of a plane graph G is a labeling of its edges with integers 1, . . . , k such that every facial trail connecting two edges with the same label contains an edge with a greater label. The smallest integer k such that G has a facial edge k-ranking is denoted by χfr′ (G). We prove that χfr′ (G) = O(log ∆∗ ) for 3-edge-connected plane graphs, where ∆∗ is the maximum face size of G. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Consider a simple graph G. Let V (G) and E (G) denote the vertex set and the edge set of G, respectively. An edge k-ranking of G is a function c : E (G) → {1, . . . , k} such that each path connecting two edges x, y satisfying c (x) = c (y) contains an edge z such that c (z ) > c (x). The smallest integer k such that G admits an edge k-ranking is denoted by χr′ (G). The edge ranking problem is to find χr′ (G) of given graph G. This problem has applications in the parallel assembly of modular products from their components [4,8] or in the parallel database query processing [5,13]. The edge ranking problem is known to be NP-hard for general graphs [10]. Some polynomial time algorithms have been developed for a few special graphs, e.g., trees [3], 2-connected outerplanar graphs [14], complete graphs [1]. In this paper we are investigating a relaxation of the edge ranking problem for the case of plane graphs, where the constraints are given by faces. There are several very recent papers that study different types of colorings/labelings of plane graphs where constraints on colorings/labelings are given by faces (see [9,11,12,16] and references therein). These papers also motivated us to introduce our new problem. A graph which can be embedded in the plane is called planar graph; a fixed embedding of a planar graph is called plane graph. Let G be a connected plane graph. Let f be a face of G of size k having boundary walk v0 , e0 , v1 , . . . , vk−1 , ek−1 , vk = v0 with vi ∈ V (G), ei ∈ E (G) and ei = vi vi+1 for every i = 0, 1, . . . , k − 1 (for the definition of the boundary walk see [7, p. 101]). A facial trail of f is any trail of the form vm , em , vm+1 , . . . , vn−1 , en−1 , vn (indices modulo k), where vi and ei are vertices and edges of the boundary walk of f . A facial trail in G is any facial trail of some face f . In this paper we consider the facial edge ranking problem of plane graphs which can be considered as a relaxation of the edge ranking problem. We focus on facial trails of plane graphs. A facial edge k-ranking of a plane graph G is a labeling of its edges with integers 1, . . . , k such that every facial trail connecting two edges with the same label contains an edge with a greater label. The smallest integer k such that G has a facial edge k-ranking is denoted by χfr′ (G). The number χfr′ (G)
∗
Corresponding author. E-mail addresses: [email protected] (J. Czap), [email protected] (S. Jendrol’).
http://dx.doi.org/10.1016/j.dam.2015.05.011 0166-218X/© 2015 Elsevier B.V. All rights reserved.
J. Czap, S. Jendrol’ / Discrete Applied Mathematics 194 (2015) 60–64
61
is called facial ranking index of G. Observe that this labeling need not be proper in a usual way. We require only that faceadjacent edges (consecutive edges of a facial trail of some face) must receive different labels. On the other hand these types of labelings coincide in the class of paths and cycles. For plane triangulations the facial edge ranking problem is equivalent to the four color problem, see e.g. the book of Saaty and Kainen [17]. From the Four Color Theorem the following result follows, see [17, p. 103]. Theorem 1. The edges of any plane triangulation T can be colored with 3 colors so that the edges bounding every face are colored distinctly, i.e.
χfr′ (T ) = 3. Shannon [19] proved that every multigraph G with maximum degree ∆ has a proper edge coloring with at most colors. This result can be reformulated for the family of plane graphs in the following way.
3 ∆ 2
Theorem 2. Let G be a 2-edge-connected plane graph with maximum face size ∆∗ . Then the edges of G can be colored with at most 23 ∆∗ colors in such a way that the edges bounding every face of G are colored distinctly. Corollary 1. If G is a 2-edge-connected plane graph with maximum face size ∆∗ , then
χfr′ (G) ≤
3 2
∆∗ .
Vizing [21,20] proved that simple planar graphs with maximum degree at least eight have the chromatic index (edge chromatic number) equal to their maximum degree. He conjectured the same if the maximum degree is either seven or six. The first part of this conjecture was proved by Sanders and Zhao [18]. Note that (also by Vizing) every graph with maximum degree ∆ has the chromatic index equal to ∆ or ∆ + 1. These results of Sanders and Zhao and of Vizing can be reformulated in the following way. Theorem 3. Let G be a 3-edge-connected plane graph with maximum face size ∆∗ ≥ 7. Then the edges of G can be colored with ∆∗ colors in such a way that the edges bounding every face of G are colored distinctly. Corollary 2. If G is a 3-edge-connected plane graph with maximum face size ∆∗ ≥ 7, then
χfr′ (G) ≤ ∆∗ . Bruoth and Horňák [2] determined the value of χr′ (Cn ) for any cycle Cn . Theorem 4 ([2]). Let Cn be a cycle on n ≥ 3 edges. Then
χr′ (Cn ) = ⌊log2 (n − 1)⌋ + 2. Corollary 3. If G is a 2-edge-connected plane graph with maximum face size ∆∗ , then
χfr′ (G) ≥ log2 (∆∗ − 1) + 2. In this paper, we show that χfr′ (G) = O(log ∆∗ ) for every 3-edge-connected plane graph G, where ∆∗ is the maximum face size of G. 2. Results Note that the facial ranking index depends on the embedding of the graph G. For example, the graph depicted in Fig. 1 with the embedding on the left has no facial edge 4-ranking (since the boundary of the outer face is a 9-cycle and from Theorem 4 it follows that χr′ (C9 ) = 5); whereas with the embedding on the right, it has a facial edge 4-ranking. The dual G∗ of a plane graph G can be obtained as follows: Corresponding to each face f of G there is a vertex f ∗ of G∗ , and corresponding to each edge e of G there is an edge e∗ of G∗ ; two vertices f ∗ and g ∗ are joined by the edge e∗ in G∗ if and only if their corresponding faces f and g are separated by the edge e in G (an edge separates the faces incident with it). 2.1. 3-connected plane graphs We write v ∈ f if a vertex v is incident with a face f . Two distinct faces f and g touch each other, if there is a vertex v such that v ∈ f and v ∈ g.
62
J. Czap, S. Jendrol’ / Discrete Applied Mathematics 194 (2015) 60–64
Fig. 1. Two embeddings of the same graph with different facial ranking indices.
Theorem 5. Let G be a 3-connected plane graph with maximum face size ∆∗ in which no two faces of size at least k, k ≥ 4, touch each other. Then log2 (∆∗ − 1) + 2 ≤ χfr′ (G) ≤ log2 (∆∗ − 1) + 2 + k.
Proof. The lower bound follows from Corollary 3. We say that a face is small if its size is at most k − 1 and big otherwise. Let B = { f1 , . . . , fℓ } be the set of big faces of G and let di denote the size of the face fi . Let the face fi be incident with the vertices vi,1 , . . . , vi,di , i ∈ {1, . . . , ℓ}. We insert the diagonals vi,1 vi,m , m ∈ {3, . . . , di − 1}, to the face fi for i ∈ {1, . . . , ℓ}. In such a way we obtain a new graph H. Observe, that every face of H has size at most k − 1. Therefore, its dual H ∗ has maximum degree at most k − 1. The graph H is 3-connected since G is 3-connected. Consequently, the dual H ∗ is simple. From Vizing’s theorem it follows that H ∗ has a proper edge labeling with at most k labels. For this labeling we use labels from the set {⌊log2 (∆∗ − 1)⌋ + 3, . . . , ⌊log2 (∆∗ − 1)⌋ + 2 + k}. This labeling induces a labeling of H in a natural way (every edge of H receives the label of the corresponding edge in H ∗ ). Clearly, the edges bounding every face of H are labeled distinctly. Now we remove the added edges from H and relabel the edges incident with big faces in such a way that we get a facial edge ranking of G. Since the maximum face size of G is ∆∗ each fi has an edge ranking with at most ⌊log2 (∆∗ − 1)⌋+ 2 labels, see Theorem 4. For this labeling we use labels from the set {1, . . . , ⌊log2 (∆∗ − 1)⌋ + 2}. Observe that no two face-adjacent edges of a small face were relabeled, since otherwise G contains two big faces which touch each other. Therefore, if a facial trail of a small face connects two edges with the same label, then this trail contains an edge which was not relabeled, hence it has a greater label. Note that the upper bound in Theorem 5 can be improved by one if k ≥ 8 (by applying Theorem 3). 2.2. 3-edge-connected plane graphs In the next part of the paper we will investigate facial edge k-ranking of 3-edge-connected plane graphs. Observe that if G is a 3-edge-connected plane graph, then its dual G∗ is a simple plane graph. Therefore, we may apply structural properties of planar graphs on the dual graph. The arboricity of a graph is the minimum number of forests into which its edges can be decomposed. Nash-Williams [15] characterized the arboricity in the following way. Theorem 6 ([15]). Let G be a simple graph. Then the arboricity of G equals
max
H ⊆G,|V (H )|≥2
|E (H )| , |V (H )| − 1
where the maximum is taken over all connected subgraphs H on at least two vertices. Since any planar graph on n vertices has at most 3n − 6 edges, from Theorem 6 it follows that the edge set of any planar graph can be decomposed into at most three forests. Gonçalves [6] improved this result for planar graphs. Theorem 7 ([6]). Every planar graph has an edge partition into at most three forests, one having maximum degree at most four. We say that an edge labeling of a plane graph G is facially proper if no two face-adjacent edges of G receive the same label. We say that a facially proper edge labeling of a plane graph is suitable if the following holds for every vertex: Let v be a vertex of degree k and let e1 , . . . , ek be the edges incident with v , listed in their clockwise order around v . If two edges ei and ej , i < j, have the same label, then there are other two edges ek and eℓ with greater labels such that k ∈ (i, j) ∩ Z (interval of integers) and ℓ ̸∈ (i, j) ∩ Z. In other words, the edges ek and eℓ are between ei and ej . Observation 1. A 2-edge-connected plane graph G has a facial edge k-ranking if and only if its dual G∗ has a suitable edge labeling with integers 1, . . . , k. This observation will play a major role in proofs below. Instead of facial edge k-ranking of G we shall investigate suitable edge labeling of G∗ .
J. Czap, S. Jendrol’ / Discrete Applied Mathematics 194 (2015) 60–64
63
Lemma 1. Let T be a tree with maximum degree ∆ ≥ 2. Then it has a suitable edge labeling with ⌊log2 (∆ − 1)⌋ + 2 labels. Moreover, this bound is tight. Proof. An edge χr′ (Pn )-ranking of a path Pn = e1 , e2 , . . . , en can be constructed by labeling ei with x + 1, where 2x is the largest power of 2 that divides i, see [1]. Clearly, the largest number in a such labeling is ⌊log2 n⌋ + 1. Therefore, an edge χr′ (Cn )-ranking of a cycle Cn = e1 , e2 , . . . , en can be constructed by labeling ei with x + 1, where 2x is the largest power of 2 that divides i for i < n and labeling the edge en with ⌊log2 (n − 1)⌋ + 2. First assume that T is a star. Let e1 , e2 , . . . , e∆ be the edges of T listed in their clockwise order around the central vertex of T . In this case it suffices to label the edges of T in the same way as it was described above for the cycle Cn . If T is not a star, then pick any vertex of T of degree ∆ to be the root. We label the edges of T starting from the root to the leaves. In each step it is sufficient to find a suitable edge labeling of a star with (at most) one prelabeled edge. Let Sk be a star on k edges. Assume that the edge e of Sk has already been labeled with label x. If k ≤ 2x−1 , then let e1 , e2 , . . . , ek = e be the edges incident with a central vertex v of Sk , listed in their clockwise order around v . In this case we label the edges of Sk (except for ek ) as it was defined for stars. Now assume that k > 2x−1 . Let j be the smallest such index of the edge of the cycle Ck = e1 , e2 , . . . , ek which receive the label x by the labeling described for the cycles. Let e1 , . . . , ej = e, . . . , ek be the edges incident with a central vertex v of Sk , listed in their clockwise order around v . It is sufficient to label the edges of Sk as it was defined for stars. Corollary 4. Let F be a forest with maximum degree ∆ ≥ 2. Then it has a suitable edge labeling with ⌊log2 (∆ − 1)⌋ + 2 labels. Proof. Suitable edge labelings of the components induce a required labeling of the whole forest.
Theorem 8. Let G be a 3-edge-connected plane graph with maximum face size ∆∗ . Then
log2 (∆∗ − 1) + 2 ≤ χfr′ (G) ≤ 2 log2 (∆∗ − 1) + 7.
Proof. The lower bound follows from Corollary 3. Let G∗ be a dual of G. The graph G∗ is simple, since G is 3-edge-connected. Theorem 7 implies that G∗ has an edge decomposition F1 ∪ F2 ∪ F3 into (at most) three forests, one having maximum degree at most four, say F3 . Let ∆(Fi ) denote the maximum degree of Fi , i = 1, 2, and assume that ∆(F1 ) ≥ ∆(F2 ). Corollary 4 implies that Fi has a suitable edge labeling with at most ⌊log2 (∆(Fi ) − 1)⌋ + 2 labels for i = 1, 2 and F3 has a such labeling with at most three labels. Let ϕ1 be a suitable edge labeling of F1 with labels 1, . . . , ⌊log2 (∆(F1 ) − 1)⌋ + 2. Let ϕ2 be a suitable edge labeling of F2 with labels ⌊log2 (∆(F1 ) − 1)⌋ + 3, . . . , 2 ⌊log2 (∆(F1 ) − 1)⌋ + 4 and let ϕ3 be a suitable edge labeling of F3 with labels 2 ⌊log2 (∆(F1 ) − 1)⌋ + 5, 2 ⌊log2 (∆(F1 ) − 1)⌋ + 6, 2 ⌊log2 (∆(F1 ) − 1)⌋ + 7. These edge labelings together induce an edge labeling of G in a natural way. Clearly, this labeling uses at most 2 ⌊log2 (∆∗ − 1)⌋ + 7 labels . So it is sufficient to show that this labeling is facial edge ranking. Assume that the edges e1 and e2 have the same label and they are incident with the same face of G. From the definition of the dual graph it follows that the corresponding edges e∗1 and e∗2 are adjacent in G∗ , moreover they belong to the same forest, say F1 , since they are labeled with the same label. The labeling ϕ1 of F1 is suitable, therefore there are edges between e∗1 and e∗2 which have greater labels. These edges correspond to edges of G which are on the two facial trails connecting e1 and e2 . We strongly believe that the following holds. Conjecture 1. There is a constant K such that for every 3-edge-connected plane graph G with maximum face size ∆∗ it holds:
χfr′ (G) ≤ log2 (∆∗ − 1) + K . 2.3. 2-connected plane graphs Theorem 9. Let G be a 2-connected plane graph with maximum face size ∆∗ in which no two faces of size at least k + 1, k ≥ 3, touch each other. Then
log2 (∆∗ − 1) + 2 ≤ χfr′ (G) ≤ log2 (∆∗ − 1) + 2 +
3 2
k.
Proof. The lower bound follows from Corollary 3. Let H be a plane graph obtained from G by triangulating all faces of size at least k + 1. Clearly, every face of H has size at most k. Therefore, its dual H ∗ has maximum degree at most k. Note that H ∗ can contain multiple edges. According to Theorem 2 the (multi)graph H ∗ has a proper edge labeling with at most 32 k labels. For this labeling we use labels from the set {⌊log2 (∆∗ − 1)⌋ + 3, . . . , ⌊log2 (∆∗ − 1)⌋ + 2 + 23 k}. This labeling induces a labeling of H in a natural way. Clearly, the edges bounding every face of H are labeled distinctly.
64
J. Czap, S. Jendrol’ / Discrete Applied Mathematics 194 (2015) 60–64
Now we remove the added edges from H and relabel the edges incident with the faces of size at least k + 1. Since the maximum face size of G is ∆∗ each such face has a facial edge ranking with at most ⌊log2 (∆∗ − 1)⌋+ 2 labels, see Theorem 4. For this labeling we use labels from the set {1, . . . , ⌊log2 (∆∗ − 1)⌋ + 2}. Now assume that a facial trail of a face f connects two edges with the same label. If the size of f is at least k + 1, then this trail contains an edge with a greater label, since the relabeling of edges incident with such faces was facial edge ranking. If the size of f is at most k, then either this trail contains an edge which was not relabeled (therefore it has a greater label) or it is also a subtrail of the adjacent face of size at least k + 1. Acknowledgments Second author’s work was supported by the Slovak Science and Technology Assistance Agency under the contract No. APVV-0023-10 and by the Slovak VEGA Grant 1/0652/12. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
H. Bodlaender, J. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Müller, Zs. Tuza, Rankings of graphs, SIAM J. Discrete Math. 11 (1998) 168–181. E. Bruoth, M. Horňák, On-line ranking number for cycles and paths, Discuss. Math. Graph Theory 19 (1999) 175–197. P. de la Torre, R. Greenlaw, A.A. Schäffer, Optimal edge ranking of trees in polynomial time, Algorithmica 13 (2005) 592–618. D. Dereniowski, Edge ranking of weighted trees, Discrete Appl. Math. 154 (2006) 1198–1209. D. Dereniowski, M. Kubale, Efficient parallel query processing by graph ranking, Fundam. Inform. 69 (2006) 273–285. D. Gonçalves, Covering planar graphs with forests, one having bounded maximum degree, J. Combin. Theory Ser. B 99 (2009) 314–322. J.L. Gross, T.W. Tucker, Topological Graph Theory, Dover, 2001. A.V. Iyer, H.D. Ratliff, G. Vijayan, Parallel Assembly of Modular Products—an Analysis, Tech. Report 88–06, Georgia Institute of Technology, 1988. S. Jendrol’, J. Miškuf, R. Soták, E. Škrabul’áková, Rainbow faces in edge-colored plane graphs, J. Graph Theory 62 (2009) 84–99. T.W. Lam, F.L. Yue, Edge ranking of graphs is hard, Discrete Appl. Math. 85 (1998) 71–86. B. Lužar, M. Mockovčiaková, R. Soták, R. Škrekovski, P. Šugerek, ℓ-facial edge colorings of graphs, Discrete Appl. Math. 181 (2015) 193–200. B. Lužar, R. Škrekovski, Improved bound on facial parity edge coloring, Discrete Math. 313 (2013) 2218–2222. K. Makino, Y. Uno, T. Ibaraki, On minimum edge ranking spanning trees, J. Algorithms 38 (2001) 411–437. S. Nakayama, S. Masuyama, A polynomial time algorithm for obtaining minimum edge ranking on two-connected outerplanar graphs, Inform. Process. Lett. 103 (2007) 216–221. C.St.J.A. Nash-Williams, Decomposition of finite graphs into forests, J. Lond. Math. Soc. 39 (1964) 12. J. Przybyło, On the facial Thue choice index via entropy compression, J. Graph Theory 77 (2014) 180–189. T.L. Saaty, P.C. Kainen, The Four-Color Problem, Assaults and Conquest, McGraw-Hill, London, 1977. D.P. Sanders, Y. Zhao, Planar graphs of maximum degree seven are class I, J. Combin. Theory Ser. B 83 (2001) 201–212. C.E. Shannon, A theorem on coloring the lines of a network, J. Math. Phys. 28 (1949) 148–151. V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz 3 (1964) 25–30. V.G. Vizing, Critical graphs with given chromatic class, Diskret. Analiz 5 (1965) 9–17.
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Facial edge ranking of plane graphs Július Czap a,∗ , Stanislav Jendrol’ b a
Department of Applied Mathematics and Business Informatics, Faculty of Economics, Technical University of Košice, Němcovej 32, 040 01 Košice, Slovakia b Institute of Mathematics, P. J. Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
article
info
Article history: Received 22 May 2014 Received in revised form 13 March 2015 Accepted 5 May 2015 Available online 27 May 2015 Keywords: Facial edge ranking Plane graph
abstract A facial edge k-ranking of a plane graph G is a labeling of its edges with integers 1, . . . , k such that every facial trail connecting two edges with the same label contains an edge with a greater label. The smallest integer k such that G has a facial edge k-ranking is denoted by χfr′ (G). We prove that χfr′ (G) = O(log ∆∗ ) for 3-edge-connected plane graphs, where ∆∗ is the maximum face size of G. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Consider a simple graph G. Let V (G) and E (G) denote the vertex set and the edge set of G, respectively. An edge k-ranking of G is a function c : E (G) → {1, . . . , k} such that each path connecting two edges x, y satisfying c (x) = c (y) contains an edge z such that c (z ) > c (x). The smallest integer k such that G admits an edge k-ranking is denoted by χr′ (G). The edge ranking problem is to find χr′ (G) of given graph G. This problem has applications in the parallel assembly of modular products from their components [4,8] or in the parallel database query processing [5,13]. The edge ranking problem is known to be NP-hard for general graphs [10]. Some polynomial time algorithms have been developed for a few special graphs, e.g., trees [3], 2-connected outerplanar graphs [14], complete graphs [1]. In this paper we are investigating a relaxation of the edge ranking problem for the case of plane graphs, where the constraints are given by faces. There are several very recent papers that study different types of colorings/labelings of plane graphs where constraints on colorings/labelings are given by faces (see [9,11,12,16] and references therein). These papers also motivated us to introduce our new problem. A graph which can be embedded in the plane is called planar graph; a fixed embedding of a planar graph is called plane graph. Let G be a connected plane graph. Let f be a face of G of size k having boundary walk v0 , e0 , v1 , . . . , vk−1 , ek−1 , vk = v0 with vi ∈ V (G), ei ∈ E (G) and ei = vi vi+1 for every i = 0, 1, . . . , k − 1 (for the definition of the boundary walk see [7, p. 101]). A facial trail of f is any trail of the form vm , em , vm+1 , . . . , vn−1 , en−1 , vn (indices modulo k), where vi and ei are vertices and edges of the boundary walk of f . A facial trail in G is any facial trail of some face f . In this paper we consider the facial edge ranking problem of plane graphs which can be considered as a relaxation of the edge ranking problem. We focus on facial trails of plane graphs. A facial edge k-ranking of a plane graph G is a labeling of its edges with integers 1, . . . , k such that every facial trail connecting two edges with the same label contains an edge with a greater label. The smallest integer k such that G has a facial edge k-ranking is denoted by χfr′ (G). The number χfr′ (G)
∗
Corresponding author. E-mail addresses: [email protected] (J. Czap), [email protected] (S. Jendrol’).
http://dx.doi.org/10.1016/j.dam.2015.05.011 0166-218X/© 2015 Elsevier B.V. All rights reserved.
J. Czap, S. Jendrol’ / Discrete Applied Mathematics 194 (2015) 60–64
61
is called facial ranking index of G. Observe that this labeling need not be proper in a usual way. We require only that faceadjacent edges (consecutive edges of a facial trail of some face) must receive different labels. On the other hand these types of labelings coincide in the class of paths and cycles. For plane triangulations the facial edge ranking problem is equivalent to the four color problem, see e.g. the book of Saaty and Kainen [17]. From the Four Color Theorem the following result follows, see [17, p. 103]. Theorem 1. The edges of any plane triangulation T can be colored with 3 colors so that the edges bounding every face are colored distinctly, i.e.
χfr′ (T ) = 3. Shannon [19] proved that every multigraph G with maximum degree ∆ has a proper edge coloring with at most colors. This result can be reformulated for the family of plane graphs in the following way.
3 ∆ 2
Theorem 2. Let G be a 2-edge-connected plane graph with maximum face size ∆∗ . Then the edges of G can be colored with at most 23 ∆∗ colors in such a way that the edges bounding every face of G are colored distinctly. Corollary 1. If G is a 2-edge-connected plane graph with maximum face size ∆∗ , then
χfr′ (G) ≤
3 2
∆∗ .
Vizing [21,20] proved that simple planar graphs with maximum degree at least eight have the chromatic index (edge chromatic number) equal to their maximum degree. He conjectured the same if the maximum degree is either seven or six. The first part of this conjecture was proved by Sanders and Zhao [18]. Note that (also by Vizing) every graph with maximum degree ∆ has the chromatic index equal to ∆ or ∆ + 1. These results of Sanders and Zhao and of Vizing can be reformulated in the following way. Theorem 3. Let G be a 3-edge-connected plane graph with maximum face size ∆∗ ≥ 7. Then the edges of G can be colored with ∆∗ colors in such a way that the edges bounding every face of G are colored distinctly. Corollary 2. If G is a 3-edge-connected plane graph with maximum face size ∆∗ ≥ 7, then
χfr′ (G) ≤ ∆∗ . Bruoth and Horňák [2] determined the value of χr′ (Cn ) for any cycle Cn . Theorem 4 ([2]). Let Cn be a cycle on n ≥ 3 edges. Then
χr′ (Cn ) = ⌊log2 (n − 1)⌋ + 2. Corollary 3. If G is a 2-edge-connected plane graph with maximum face size ∆∗ , then
χfr′ (G) ≥ log2 (∆∗ − 1) + 2. In this paper, we show that χfr′ (G) = O(log ∆∗ ) for every 3-edge-connected plane graph G, where ∆∗ is the maximum face size of G. 2. Results Note that the facial ranking index depends on the embedding of the graph G. For example, the graph depicted in Fig. 1 with the embedding on the left has no facial edge 4-ranking (since the boundary of the outer face is a 9-cycle and from Theorem 4 it follows that χr′ (C9 ) = 5); whereas with the embedding on the right, it has a facial edge 4-ranking. The dual G∗ of a plane graph G can be obtained as follows: Corresponding to each face f of G there is a vertex f ∗ of G∗ , and corresponding to each edge e of G there is an edge e∗ of G∗ ; two vertices f ∗ and g ∗ are joined by the edge e∗ in G∗ if and only if their corresponding faces f and g are separated by the edge e in G (an edge separates the faces incident with it). 2.1. 3-connected plane graphs We write v ∈ f if a vertex v is incident with a face f . Two distinct faces f and g touch each other, if there is a vertex v such that v ∈ f and v ∈ g.
62
J. Czap, S. Jendrol’ / Discrete Applied Mathematics 194 (2015) 60–64
Fig. 1. Two embeddings of the same graph with different facial ranking indices.
Theorem 5. Let G be a 3-connected plane graph with maximum face size ∆∗ in which no two faces of size at least k, k ≥ 4, touch each other. Then log2 (∆∗ − 1) + 2 ≤ χfr′ (G) ≤ log2 (∆∗ − 1) + 2 + k.
Proof. The lower bound follows from Corollary 3. We say that a face is small if its size is at most k − 1 and big otherwise. Let B = { f1 , . . . , fℓ } be the set of big faces of G and let di denote the size of the face fi . Let the face fi be incident with the vertices vi,1 , . . . , vi,di , i ∈ {1, . . . , ℓ}. We insert the diagonals vi,1 vi,m , m ∈ {3, . . . , di − 1}, to the face fi for i ∈ {1, . . . , ℓ}. In such a way we obtain a new graph H. Observe, that every face of H has size at most k − 1. Therefore, its dual H ∗ has maximum degree at most k − 1. The graph H is 3-connected since G is 3-connected. Consequently, the dual H ∗ is simple. From Vizing’s theorem it follows that H ∗ has a proper edge labeling with at most k labels. For this labeling we use labels from the set {⌊log2 (∆∗ − 1)⌋ + 3, . . . , ⌊log2 (∆∗ − 1)⌋ + 2 + k}. This labeling induces a labeling of H in a natural way (every edge of H receives the label of the corresponding edge in H ∗ ). Clearly, the edges bounding every face of H are labeled distinctly. Now we remove the added edges from H and relabel the edges incident with big faces in such a way that we get a facial edge ranking of G. Since the maximum face size of G is ∆∗ each fi has an edge ranking with at most ⌊log2 (∆∗ − 1)⌋+ 2 labels, see Theorem 4. For this labeling we use labels from the set {1, . . . , ⌊log2 (∆∗ − 1)⌋ + 2}. Observe that no two face-adjacent edges of a small face were relabeled, since otherwise G contains two big faces which touch each other. Therefore, if a facial trail of a small face connects two edges with the same label, then this trail contains an edge which was not relabeled, hence it has a greater label. Note that the upper bound in Theorem 5 can be improved by one if k ≥ 8 (by applying Theorem 3). 2.2. 3-edge-connected plane graphs In the next part of the paper we will investigate facial edge k-ranking of 3-edge-connected plane graphs. Observe that if G is a 3-edge-connected plane graph, then its dual G∗ is a simple plane graph. Therefore, we may apply structural properties of planar graphs on the dual graph. The arboricity of a graph is the minimum number of forests into which its edges can be decomposed. Nash-Williams [15] characterized the arboricity in the following way. Theorem 6 ([15]). Let G be a simple graph. Then the arboricity of G equals
max
H ⊆G,|V (H )|≥2
|E (H )| , |V (H )| − 1
where the maximum is taken over all connected subgraphs H on at least two vertices. Since any planar graph on n vertices has at most 3n − 6 edges, from Theorem 6 it follows that the edge set of any planar graph can be decomposed into at most three forests. Gonçalves [6] improved this result for planar graphs. Theorem 7 ([6]). Every planar graph has an edge partition into at most three forests, one having maximum degree at most four. We say that an edge labeling of a plane graph G is facially proper if no two face-adjacent edges of G receive the same label. We say that a facially proper edge labeling of a plane graph is suitable if the following holds for every vertex: Let v be a vertex of degree k and let e1 , . . . , ek be the edges incident with v , listed in their clockwise order around v . If two edges ei and ej , i < j, have the same label, then there are other two edges ek and eℓ with greater labels such that k ∈ (i, j) ∩ Z (interval of integers) and ℓ ̸∈ (i, j) ∩ Z. In other words, the edges ek and eℓ are between ei and ej . Observation 1. A 2-edge-connected plane graph G has a facial edge k-ranking if and only if its dual G∗ has a suitable edge labeling with integers 1, . . . , k. This observation will play a major role in proofs below. Instead of facial edge k-ranking of G we shall investigate suitable edge labeling of G∗ .
J. Czap, S. Jendrol’ / Discrete Applied Mathematics 194 (2015) 60–64
63
Lemma 1. Let T be a tree with maximum degree ∆ ≥ 2. Then it has a suitable edge labeling with ⌊log2 (∆ − 1)⌋ + 2 labels. Moreover, this bound is tight. Proof. An edge χr′ (Pn )-ranking of a path Pn = e1 , e2 , . . . , en can be constructed by labeling ei with x + 1, where 2x is the largest power of 2 that divides i, see [1]. Clearly, the largest number in a such labeling is ⌊log2 n⌋ + 1. Therefore, an edge χr′ (Cn )-ranking of a cycle Cn = e1 , e2 , . . . , en can be constructed by labeling ei with x + 1, where 2x is the largest power of 2 that divides i for i < n and labeling the edge en with ⌊log2 (n − 1)⌋ + 2. First assume that T is a star. Let e1 , e2 , . . . , e∆ be the edges of T listed in their clockwise order around the central vertex of T . In this case it suffices to label the edges of T in the same way as it was described above for the cycle Cn . If T is not a star, then pick any vertex of T of degree ∆ to be the root. We label the edges of T starting from the root to the leaves. In each step it is sufficient to find a suitable edge labeling of a star with (at most) one prelabeled edge. Let Sk be a star on k edges. Assume that the edge e of Sk has already been labeled with label x. If k ≤ 2x−1 , then let e1 , e2 , . . . , ek = e be the edges incident with a central vertex v of Sk , listed in their clockwise order around v . In this case we label the edges of Sk (except for ek ) as it was defined for stars. Now assume that k > 2x−1 . Let j be the smallest such index of the edge of the cycle Ck = e1 , e2 , . . . , ek which receive the label x by the labeling described for the cycles. Let e1 , . . . , ej = e, . . . , ek be the edges incident with a central vertex v of Sk , listed in their clockwise order around v . It is sufficient to label the edges of Sk as it was defined for stars. Corollary 4. Let F be a forest with maximum degree ∆ ≥ 2. Then it has a suitable edge labeling with ⌊log2 (∆ − 1)⌋ + 2 labels. Proof. Suitable edge labelings of the components induce a required labeling of the whole forest.
Theorem 8. Let G be a 3-edge-connected plane graph with maximum face size ∆∗ . Then
log2 (∆∗ − 1) + 2 ≤ χfr′ (G) ≤ 2 log2 (∆∗ − 1) + 7.
Proof. The lower bound follows from Corollary 3. Let G∗ be a dual of G. The graph G∗ is simple, since G is 3-edge-connected. Theorem 7 implies that G∗ has an edge decomposition F1 ∪ F2 ∪ F3 into (at most) three forests, one having maximum degree at most four, say F3 . Let ∆(Fi ) denote the maximum degree of Fi , i = 1, 2, and assume that ∆(F1 ) ≥ ∆(F2 ). Corollary 4 implies that Fi has a suitable edge labeling with at most ⌊log2 (∆(Fi ) − 1)⌋ + 2 labels for i = 1, 2 and F3 has a such labeling with at most three labels. Let ϕ1 be a suitable edge labeling of F1 with labels 1, . . . , ⌊log2 (∆(F1 ) − 1)⌋ + 2. Let ϕ2 be a suitable edge labeling of F2 with labels ⌊log2 (∆(F1 ) − 1)⌋ + 3, . . . , 2 ⌊log2 (∆(F1 ) − 1)⌋ + 4 and let ϕ3 be a suitable edge labeling of F3 with labels 2 ⌊log2 (∆(F1 ) − 1)⌋ + 5, 2 ⌊log2 (∆(F1 ) − 1)⌋ + 6, 2 ⌊log2 (∆(F1 ) − 1)⌋ + 7. These edge labelings together induce an edge labeling of G in a natural way. Clearly, this labeling uses at most 2 ⌊log2 (∆∗ − 1)⌋ + 7 labels . So it is sufficient to show that this labeling is facial edge ranking. Assume that the edges e1 and e2 have the same label and they are incident with the same face of G. From the definition of the dual graph it follows that the corresponding edges e∗1 and e∗2 are adjacent in G∗ , moreover they belong to the same forest, say F1 , since they are labeled with the same label. The labeling ϕ1 of F1 is suitable, therefore there are edges between e∗1 and e∗2 which have greater labels. These edges correspond to edges of G which are on the two facial trails connecting e1 and e2 . We strongly believe that the following holds. Conjecture 1. There is a constant K such that for every 3-edge-connected plane graph G with maximum face size ∆∗ it holds:
χfr′ (G) ≤ log2 (∆∗ − 1) + K . 2.3. 2-connected plane graphs Theorem 9. Let G be a 2-connected plane graph with maximum face size ∆∗ in which no two faces of size at least k + 1, k ≥ 3, touch each other. Then
log2 (∆∗ − 1) + 2 ≤ χfr′ (G) ≤ log2 (∆∗ − 1) + 2 +
3 2
k.
Proof. The lower bound follows from Corollary 3. Let H be a plane graph obtained from G by triangulating all faces of size at least k + 1. Clearly, every face of H has size at most k. Therefore, its dual H ∗ has maximum degree at most k. Note that H ∗ can contain multiple edges. According to Theorem 2 the (multi)graph H ∗ has a proper edge labeling with at most 32 k labels. For this labeling we use labels from the set {⌊log2 (∆∗ − 1)⌋ + 3, . . . , ⌊log2 (∆∗ − 1)⌋ + 2 + 23 k}. This labeling induces a labeling of H in a natural way. Clearly, the edges bounding every face of H are labeled distinctly.
64
J. Czap, S. Jendrol’ / Discrete Applied Mathematics 194 (2015) 60–64
Now we remove the added edges from H and relabel the edges incident with the faces of size at least k + 1. Since the maximum face size of G is ∆∗ each such face has a facial edge ranking with at most ⌊log2 (∆∗ − 1)⌋+ 2 labels, see Theorem 4. For this labeling we use labels from the set {1, . . . , ⌊log2 (∆∗ − 1)⌋ + 2}. Now assume that a facial trail of a face f connects two edges with the same label. If the size of f is at least k + 1, then this trail contains an edge with a greater label, since the relabeling of edges incident with such faces was facial edge ranking. If the size of f is at most k, then either this trail contains an edge which was not relabeled (therefore it has a greater label) or it is also a subtrail of the adjacent face of size at least k + 1. Acknowledgments Second author’s work was supported by the Slovak Science and Technology Assistance Agency under the contract No. APVV-0023-10 and by the Slovak VEGA Grant 1/0652/12. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
H. Bodlaender, J. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Müller, Zs. Tuza, Rankings of graphs, SIAM J. Discrete Math. 11 (1998) 168–181. E. Bruoth, M. Horňák, On-line ranking number for cycles and paths, Discuss. Math. Graph Theory 19 (1999) 175–197. P. de la Torre, R. Greenlaw, A.A. Schäffer, Optimal edge ranking of trees in polynomial time, Algorithmica 13 (2005) 592–618. D. Dereniowski, Edge ranking of weighted trees, Discrete Appl. Math. 154 (2006) 1198–1209. D. Dereniowski, M. Kubale, Efficient parallel query processing by graph ranking, Fundam. Inform. 69 (2006) 273–285. D. Gonçalves, Covering planar graphs with forests, one having bounded maximum degree, J. Combin. Theory Ser. B 99 (2009) 314–322. J.L. Gross, T.W. Tucker, Topological Graph Theory, Dover, 2001. A.V. Iyer, H.D. Ratliff, G. Vijayan, Parallel Assembly of Modular Products—an Analysis, Tech. Report 88–06, Georgia Institute of Technology, 1988. S. Jendrol’, J. Miškuf, R. Soták, E. Škrabul’áková, Rainbow faces in edge-colored plane graphs, J. Graph Theory 62 (2009) 84–99. T.W. Lam, F.L. Yue, Edge ranking of graphs is hard, Discrete Appl. Math. 85 (1998) 71–86. B. Lužar, M. Mockovčiaková, R. Soták, R. Škrekovski, P. Šugerek, ℓ-facial edge colorings of graphs, Discrete Appl. Math. 181 (2015) 193–200. B. Lužar, R. Škrekovski, Improved bound on facial parity edge coloring, Discrete Math. 313 (2013) 2218–2222. K. Makino, Y. Uno, T. Ibaraki, On minimum edge ranking spanning trees, J. Algorithms 38 (2001) 411–437. S. Nakayama, S. Masuyama, A polynomial time algorithm for obtaining minimum edge ranking on two-connected outerplanar graphs, Inform. Process. Lett. 103 (2007) 216–221. C.St.J.A. Nash-Williams, Decomposition of finite graphs into forests, J. Lond. Math. Soc. 39 (1964) 12. J. Przybyło, On the facial Thue choice index via entropy compression, J. Graph Theory 77 (2014) 180–189. T.L. Saaty, P.C. Kainen, The Four-Color Problem, Assaults and Conquest, McGraw-Hill, London, 1977. D.P. Sanders, Y. Zhao, Planar graphs of maximum degree seven are class I, J. Combin. Theory Ser. B 83 (2001) 201–212. C.E. Shannon, A theorem on coloring the lines of a network, J. Math. Phys. 28 (1949) 148–151. V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz 3 (1964) 25–30. V.G. Vizing, Critical graphs with given chromatic class, Diskret. Analiz 5 (1965) 9–17.