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Coloring Decompositions of Complete Geometric Graphs

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Coloring Decompositions of Complete Geometric Graphs COLORING DECOMPOSITIONS OF COMPLETE GEOMETRIC GRAPHS Clemens Huemer1 Dolores Lara2 Christian Rubio-Montiel3;4;5 1Departament de Matem`atiques, Universitat Polit`ecnicade Catalunya, Spain. e-mail: [email protected] 2Departamento de Computaci´on,Centro de Investigaci´ony de Estudios Avanzados del Instituto Polit´ecnicoNacional, Mexico. e-mail: [email protected] 3 Divisi´onde Matem´aticase Ingenier´ıa,FES Acatl´an,Universidad Nacional Aut´onomade M´exico.e-mail: [email protected] 4UMI LAFMIA 3175 CNRS at CINVESTAV-IPN, Mexico. 5Department of Algebra, Comenius University, Slovakia. April 24, 2019 Abstract A decomposition of a non-empty simple graph G is a pair [G; P ], such that P is a set of non-empty induced subgraphs of G, and every edge of G belongs to exactly one subgraph in P . The chromatic index χ0([G; P ]) of a decomposition [G; P ] is the smallest number k for which there exists a k-coloring of the elements of P in such a way that: for every element of P all of its edges have the same color, and if two members of P share at least one vertex, then they have different colors. A long standing conjecture of Erd}os-Faber-Lov´asz states that every decomposition [Kn;P ] 0 of the complete graph Kn satisfies χ ([Kn;P ]) ≤ n. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922. 1 general position. We also consider the particular case when the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions. Keywords: geometric graphs, coloring, geometric chromatic index. 1. Introduction In 1972, Paul Erd}os,Vance Faber and L´aszl´oLov´asz[13] made the following conjecture: CONJECTURE 1.1. Let jAij = n, 1 ≤ i ≤ n, and suppose that jAi \Ajj ≤ 1, Sn for 1 ≤ i < j ≤ n, then one can color the elements of the union i=1 Ai by n colors, so that every set contains elements of all colors. This conjecture is called the Erd}os{Faber{Lov´aszconjecture (for short EFL- conjecture) which has been formulated in terms of decompositions of graphs [3, 4]. A decomposition of a non-empty simple graph G is a pair [G; P ], such that (1) P is a set of non-empty induced subgraphs of G, and (2) every edge of G belongs to exactly one subgraph in P . We can think of P as being a partition of the edges of G. A k-P -coloring (or a k-P -proper coloring) of a decomposition [G; P ] is a function that assigns to each edge of G a color from a set of k colors, so that (1) for every H 2 P all of its edges have the same color, and (2) for any two H1;H2 2 P , if the two graphs have at least one vertex in common, then 0 E(H1) and E(H2) have different colors. The chromatic index χ ([G; P ]) of a decomposition [G; P ] is the smallest number k for which there exists a k-P - coloring of [G; P ]. The authors of [3] and [4] give an equivalent formulation of the well-known EFL-conjecture in terms of the chromatic index of decompositions of the com- plete graph: CONJECTURE 1.2. Every decomposition [Kn;P ] of the complete graph Kn satisfies 0 χ ([Kn;P ]) ≤ n: Note that for every decomposition [Kn;P ] of the complete graph Kn, since P is a set of non-empty induced subgraphs of Kn, the set P consists of complete subgraphs of Kn. To avoid repetition, we do not reiterate this in what follows, but we ask the reader to keep it in mind throughout the paper. In this paper, inspired by Conjecture 1.2, we study a problem about de- compositions of complete geometric graphs. A geometric graph G1 is a drawing in the plane of a graph G, such that its vertices are points in general position (no three of them are collinear), and its edges are straight-line segments. Two 1To avoid any possible confusion, geometric graphs will be denoted by Sans Serif Math letters. 2 geometric graphs have nonempty intersection if either (1) they have a common vertex, or (2) there exists a pair of edges, one from each graph, that crosses. A complete geometric graph of order n, denoted Kn, is a graph in wich each vertex is adjacent to every other vertex. A decomposition [G; P ] of a graph G naturally induces a decomposition [G;P ] of any geometric graph G.A k-P -coloring (or a k-P -proper coloring) of a decomposition [G;P ] maps to the edges of G a color from a set with k colors, so that (1) for every H 2 P all of its edges have the same color, and (2) for any two H1; H2 2 P , if the two graphs have nonempty intersection, then E(H1) and E(H2) have different colors. The smallest positive integer k for which there is a k-P -coloring of [G;P ] is the chromatic index of a decomposition [G;P ] and it is denoted by χ0([G;P ]). In this paper we give lower and upper bounds for the chromatic index of several families of decompositions (in complete subgraphs) of complete geomet- ric graphs. Furthermore, we state a conjecture on an upper bound on χ0([G;P ]) for any partition P . The paper is organized as follows. In Section 2 we consider complete geometric graphs whose set of vertices are points in general position, in Section 3 we consider the case in which the vertices of the complete geometric graph are in convex position (they are the vertices of a convex polygon), finally, in Section 4 we state a few conjectures. 1.1 Notation, Terminology and Definitions Throughout this paper we assume that all sets of points in the plane are in general position. Note that every set of n points in the plane induces a complete geometric graph. Let S be a set of n points in the plane and let Kn be the complete geometric graph induced by S. For brevity, we refer to the points in S as vertices of Kn, and to the straight-line segments connecting two points in S as the edges of Kn. In the remainder of this subsection we present some definitions related with designs. DEFINITION 1.1. Given integers n and κ, a 2-(n; κ)-design D is a pair (P; B), where P is a set of n elements, and B is a collection of κ-subsets of P with the property that each subset of P of size 2 is a subset of exactly one member of B. The members of P are called points and the members of B are called blocks. DEFINITION 1.2. D = (Zn; B) is a cyclic design if B is fixed by the auto- morphism i 7! i + 1. When it is clear from the context, we refer to a 2-(n; κ)-design simply as 2- design. 2-designs are commonly known in the literature as BIBD, standing for balanced incomplete block design. We can think about any 2-design D as a decomposition of the complete graph. That is, D is a particular case of a decomposition of the complete graph, in which, in the natural way, every block is an element of the partition, see [16]. κn It was proven in [9] that any 2-design has chromatic index at most κ−1 , and that, every cyclic 2-design satisfies the EFL conjecture. However, in general, 3 Figure 1: The unique Steiner triple system ST S(9) as a decomposition of K9. the EFL conjecture is open for 2-designs, even for κ = 3. A 2-(n; 3)-design is called Steiner triple system, and it is denoted by ST S(n). It is well-known that ST S(n) exists if and only if n is congruent to 1 or 3 modulo 6 (see [18]). It is also well-known that a cyclic ST S(n) exists if and only if n 6= 9 and n is congruent to 1 or 3 modulo 6 (see [11, 17]). Figure 1 shows the unique ST S(9), see [10]. There is another special type of well-known 2-design: A projective plane is a 2-(q2 + q + 1; q + 1)-design having the following properties: 1. Given any two distinct blocks there is exactly one point incident with both of them. 2. There are four points so that no line is incident with more than two of them. The number q is called the order of the projective plane. A projective plane of order q is denoted by Πq. It is known [15] that there exists at least one projective plane of order q for any prime power q. 2. Points in General Position Let D be a 2-(n; 2)-design. D induces the decomposition [Kn;E(Kn)] of Kn. 0 Therefore, Vizing's theorem verifies the EFL conjecture, since χ ([Kn;E(Kn)]) 0 0 is the usual chromatic index χ (Kn) of the complete graph, and χ (Kn) ≤ n.
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