The 6-girth-thickness of the complete graph

Transcription

1 The -girth-thickness of the complete graph arxiv:0.0v [math.co] Jul 0 Héctor Castañeda-López Pablo C. Palomino Andrea B. Ramos-Tort Christian Rubio-Montiel Claudia Silva-Ruíz July 0, 0 Abstract The g-girth-thickness θ(g, G) of a graph G is the minimum number of planar subgraphs of girth at least g whose union is G. In this paper, we determine the -girththickness θ(, K n ) of the complete graph K n in almost all cases. And also, we calculate by computer the missing value of θ(, K n ). Keywords: Thickness, planar decomposition, complete graph, girth. 00 Mathematics Subject Classification: 0C0. Introduction In this paper, all graphs are finite and simple. A graph in which any two vertices are adjacent is called a complete graph and it is denoted by K n if it has n vertices. If a graph can be drawn in the Euclidean plane such that no inner point of its edges is a vertex or lies on another edge, then the graph G is called planar. The girth of a graph is the size of its shortest cycle or if it is acyclic. It is known that an acyclic graph of order n has size at most n and a planar graph of order n and finite girth g has size at most g g (n ), see []. Facultad de Ciencias, Universidad Autónoma del Estado de México, 0000, Toluca, Mexico, hcastanedal@alumno.uaemex.mx. Facultad de Ciencias, Universidad Nacional Autónoma de México, 00, Mexico City, Mexico, [pablop ramos tort callame]@ciencias.unam.mx. División de Matemáticas e Ingeniería, FES Acatlán, Universidad Nacional Autónoma de México, 0, Naucalpan, Mexico, christian@apolo.acatlan.unam.mx.

2 The thickness θ(g) of a graph G is the minimum number of planar subgraphs whose union is G. Equivalently, it is the minimum number of colors used in any edge coloring of G such that each set of edges in the same chromatic class induces a planar subgraph. The concept of the thickness was introduced by Tutte []. The problem to determine the thickness of a graph G is NP-hard [], and only a few of exact results are known, for instance, when G is a complete graph [,, ], a complete multipartite graph [,, 0] or a hypercube []. Generalizations of the thickness for the complete graphs also have been studied such that the outerthickness θ o, defined similarly but with outerplanar instead of planar [], and the S- thickness θ S, considering the thickness on a surface S instead of the plane []. The thickness has many applications, for example, in the design of circuits [], in the Ringel s earth-moon problem [], or to bound the achromatic numbers of planar graphs []. See also []. In [], the g-girth-thickness θ(g, G) of a graph G was defined as the minimum number of planar subgraphs of girth at least g whose union is G. Indeed, the g-girth thickness generalizes the thickness when g = and the arboricity number when g =. This paper is organized as follows. In Section, we obtain the -girth-thickness θ(, K n ) of the complete graph K n getting that θ(, K n ) equals n+, except for n = t +, t and n, for which θ(, K ) =. In Section, we show that there exists a set of planar triangle-free subgraphs of K 0 whose union is K 0. The decomposition was found by computer and, as a consequence, we disproved the conjecture that appears in [] about the missing case of the -girth-thickness of the complete graph. Determining θ(, K n ) A planar graph of n vertices with girth at least has size at most (n )/ for n and size at most n for n, therefore, the -girth-thickness θ(, K n ) of the complete graph K n is at least n(n ) n + = + n + = (n ) n for n, as well as, n+ for n {,,, }. We have the following theorem. Theorem.. The -girth-thickness θ(, K n ) of K n is equal to n+ except possibly when n = t +, for t, and n for which θ(, K ) =. Proof. To begin with, Figure displays equality for n =,,, 0 with θ(, K n ) =,,,, respectively. The rest of the cases for n 0 are obtained by the hereditary property of

3 a) b) c) d) Figure : A decomposition of K n into θ(, K n ) planar subgraphs of girth at least : a) for n =, b) for n =, c) for n = and d) for n = 0.

4 the induced subgraphs. We remark that the decomposition of K 0 was found by computer using the database of the connected planar graphs of order 0 that appears in []. Now, we need to distinguish two main cases, namely, when t is even or t is odd for n = t, that is, when n = k and n = k + for k. The cases n = k and n = k +, i.e., for n = t +, are obtained by the hereditary property of the induced subgraphs, that is, since K k K k and K k+ K k+, we have k + θ(, K k ) θ(, K k ) and k + θ(, K k+ ) θ(, K k+ ), respectively. Therefore, the case of n = k shows a decomposition of K k into k + planar subgraphs of girth at least, while the case of n = k+ shows a decomposition of K k+ into k+ planar subgraphs of girth at least. Both constructions are based on the planar decomposition of K k of Beineke and Harary [] (see also [,, ]) but we use the combinatorial approach given in []. Then, for the sake of completeness, we give a decomposition of K k in order to obtain its usual thickness. In the remainder of this proof, all sums are taken modulo k. We recall that complete graphs of even order k are decomposable into a cyclic factorization of Hamiltonian paths, see [0]. Let G x be a complete graph of order k, label its vertex set V (G x ) as {x, x,..., x k } and let Fi x be the Hamiltonian path with edges x i x i+, x i+ x i, x i x i+, x i+ x i,..., x i+k+ x i+k, for all i {,,..., k}. The partition {E(F x ), E(F x ),..., E(Fk x )} is such factorization of G x. We remark that the center of Fi x has the edge e x i = x i+ k x i+ k, see Figure. a) b) x i x i x i+ x i+ k x i x i x i+ x i+ k x i+ k x i+ k x i+ k x i+k x i+k+ x i+k x i+ k x i+ k + x i+ k xi+k+ x i+k x i+k Figure : The Hamiltonian path Fi x : Left a) The edge e x i edge e x i in bold for k even. in bold for k odd. Right b) The Let G u, G v and G w be the complete subgraphs of K k having k vertices each of them and such that G s a subgraph of K k \ V (G u ) and G s K k \ (V (G u ) V (G v )). The

5 vertices of V (G u ), V (G v ) and V (G w ) are labeled as {u, u,..., u k }, {v, v,..., v k } and {w, w,..., w k }, respectively. Let x be an element of {u, v, w}. Take the cyclic factorization {E(F x ), E(F x ),..., E(Fk x)} of G x into Hamiltonian paths and denote as P xi and P xi+k the subpaths of Fi x containing k vertices and the leaves x i and x i+k, respectively. We define the other leaves of P xi and P xi+k as f(x i ) and f(x i+k ), respectively and according to the parity of k, that is (see Figure ), x i+ f(x i ) = k x i+ k if k is odd, if k is even. x i+ and f(x i+k ) = k x i+ k if k is odd, if k is even. We remark that the set of edges {x i x i+k : i k} is the same set of edges that {f(x i )f(x i+k ): i k}. Now, we construct the maximal planar subgraphs G, G,...,G k and a matching G k+ with k vertices each in the following way. Let G k+ be the perfect matching with the edges u j u j+k, v j v j+k and w j w j+k for j {,,..., k}. For each i {,,..., k}, let G i be the spanning planar graph of K k whose adjacencies are given as follows: we take the paths, P ui, P ui+k, P vi, P vi+k, P wi and P wi+k and insert them in the octahedron with the vertices, +k,,, and as is shown in Figure (Left). The vertex x j of each path P xj is identified with the vertex x j in the corresponding triangle face and join all the other vertices of the path with both of the other vertices of the triangle face, see Figure (Right). +k +k G i G i P ui+k P wi P vi P vi+k P wi+k P ui Figure : (Left) The octahedron subgraph of the graph G i. (Right) The graph G i. By construction of G i, K k = k+ G i, see [, ] to check a full proof. In consequence, the k + i= planar subgraphs G i show that θ(, K k ) k + and then, θ(, K k ) = k + owing to the

6 ( k ) fact that θ(, K k ) = k +. (k ) Now, we proceed to prove that θ(, K k ) k + in Case and θ(, K k+ ) k + in Case. The main idea of both cases is divide each G i into two subgraphs of girth for any i {,..., k}.. Case n = k. Consider the set of planar subgraphs {G, G,..., G k+ } of K k which is described above. Step. For each i {,..., k}, remove the six edges of the triangles and +k. Step. For each i {,..., k}, divide the obtained subgraph into two subgraphs Hi and Hi as follows: The maximum matching of P xi incident to the vertex f(x i ) belongs to Hi (see dotted subgraph in Figure ) while the maximum matching of P xi+k incident to the vertex f(x i+k ) belongs to Hi. Next, the rest of the edges joined to the vertices of the paths P xi and P xi+k, in an alternative way from the exterior region to the region with the vertices {,, }, belong to Hi and Hi respectively, such that the edges f( )+k, f( ) and f( ) belong to Hi and the edges f( ), f( )+k and f( ) belong to Hi, see Figure. +k f(u f( ) i+k ) f( ) f( ) f( ) f( ) Figure : Partial modification of the subgraph G i. Step. Consider the removed edges in Step, add the edges f( )f(+k ) and f(+k )f( ) to Hi and the edges f( )f( ) and f( )f( ) to Hi, see Figure. The rest of the edges removed in Step are added to G k+ getting the subgraph H k+ which is the union of the paths {f( ), f( ), f( ), f( ), f( ), f(+k )}.. Case n = k +.

7 +k H i H i +k f(+k ) f( ) f( ) f( ) f(+k ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) Figure : Subgraphs H i and H i for the Case. Consider the set of planar subgraphs {G, G,..., G k+ } of K k which is described above as well as Step and of the previous case. Step. Add three vertices u, v and n the subgraphs Hi and Hi, for each i {,..., k}, and the edges u, uf( ), v, vf( ), w, wf(+k ) into Hi as well as the edges u, uf( ), v+k, vf( ), w, wf( ) into Hi, see Figure. +k +k v f(+k ) f( ) f( ) u f( ) w v f( ) f( ) f(+k ) f( ) f( ) f( ) u f( ) w f( ) Figure : Subgraphs H i and H i for Case. Step. On one hand, remains to define the adjacencies between u, v, w and all the adjacencies between u and, v and, w and, for each j {,..., k}. On the other hand, the edges of the graph G k+ together with the removed edges of the Step form a set of triangle prisms which we split into two subgraphs called H k+ and H k+ in the following way: a) The adjacency vs in H k+ while the adjacencies uv and uw are in H k+, see Figure.

8 b) The set of adjacencies vv j+k, ww j, ww j+k and uu j+k are in Hk+ while the set of adjacencies vv j, and uu j are in Hk+, for each j {,..., k}, see Figure. c) The subgraph Hk+ contains the adjacencies v j+kv j, v j u j, u j w j and w j+k u j+k (a set of subgraphs P K ) and the subgraph Hk+ contains the adjacencies u ju j+k, u j+k v j+k, v j+k w j+k,w j+k w j and w j v j (a set of subgraphs P ) for all j {,..., k}, see Figure. u j+k u j+k Hk+ Hk+ u u w u j u j v v j w j v j w j v j+k w j+k v j+k w j+k v w Figure : Partial subgraphs H k+ and H k+. By the small cases and the two main cases, the theorem follows. The -girth thickness of K 0 In [], Rubio-Montiel gave a decomposition of K n into θ(, K n ) = n+ triangle-free planar subgraphs, except for n = 0. In that case, it was bounded by θ(, K 0 ) and conjectured that the correct value was the upper bound. Using the database of the connected planar graphs of order 0 that appears in [] and the SageMath program, we found two decompositions of K 0 into planar subgraphs of girth at least illustrated in Figure. In summary, the correct value of θ(, K n ) was the lower bound and then, we have the following theorem. Theorem.. The -girth-thickness θ(, K n ) of K n equals n+ for n and θ(, K ) =.

9 0 0 0, Figure : Two planar decompositions of K 0 into three subgraphs of girth. Acknowledgments Part of the work was done during the IV Taller de Matemáticas Discretas, held at Campus- Juriquilla, Universidad Nacional Autónoma de México, Querétaro City, Mexico on June, 0. We thank Miguel Raggi and Jessica Sánchez for their useful discussions and help with the SageMath program. References [] A. Aggarwal, M. Klawe and P. Shor, Multilayer grid embeddings for VLSI, Algorithmica (), no.,. [] V. B. Alekseev and V. S. Gončakov, The thickness of an arbitrary complete graph, Mat. Sb. (N.S.) 0() (), no., 0. [] G. Araujo-Pardo, F. E. Contreras-Mendoza, S. J. Murillo-García, A. B. Ramos-Tort and C. Rubio-Montiel, Complete colorings of planar graphs, preprint arxiv:0.00 (0). [] L. W. Beineke, Minimal decompositions of complete graphs into subgraphs with embeddability properties, Canad. J. Math. (), 000. [] L. W. Beineke and F. Harary, On the thickness of the complete graph, Bull. Amer. Math. Soc. 0 (), 0.

10 [] L. W. Beineke and F. Harary, The thickness of the complete graph, Canad. J. Math. (), 0. [] L. W. Beineke, F. Harary and J. W. Moon, On the thickness of the complete bipartite graph, Proc. Cambridge Philos. Soc. 0 (),. [] J.A. Bondy and U.S.R. Murty, Graph theory, Graduate Texts in Mathematics, vol., Springer, New York, 00. [] G. Brinkmann, K. Coolsaet, J. Goedgebeur and H. Mélot, House of Graphs: a database of interesting graphs, Discrete Appl. Math. (0), no. -,, Available at Accessed: [0] G. Chartrand and P. Zhang, Chromatic graph theory, Discrete Mathematics and its Applications, CRC Press, 00. [] R. K. Guy and R. J. Nowakowski, The outerthickness & outercoarseness of graphs. I. The complete graph & the n-cube, Topics in combinatorics and graph theory (Oberwolfach, 0), Physica, Heidelberg, 0, 0. [] B. Jackson and G. Ringel, Variations on Ringel s earth-moon problem, Discrete Math. (000), no. -,. [] M. Kleinert, Die Dicke des n-dimensionalen Würfel-Graphen, J. Combin. Theory (), 0. [] A. Mansfield, Determining the thickness of graphs is NP-hard, Math. Proc. Cambridge Philos. Soc. (), no.,. [] P. Mutzel, Odenthal T. and M. Scharbrodt, The thickness of graphs: a survey, Graphs Combin. (), no.,. [] C. Rubio-Montiel, The -girth-thickness of the complete graph, Ars Math. Contemp. (0), no.,. [] W. T. Tutte, The thickness of a graph, Indag. Math. (),. [] J. M. Vasak, The thickness of the complete graphs, ProQuest LLC, Ann Arbor, MI,, Thesis (Ph.D.) University of Illinois at Urbana-Champaign. [] Y. Yang, A note on the thickness of K l,m,n, Ars Combin. (0),. [0] Y. Yang, Remarks on the thickness of K n,n,n, Ars Math. Contemp. (0), no.,. 0