Chromatic Numbers of Simplicial Manifolds

Higher chromatic numbers $\chi_s$ of simplicial complexes naturally generalize the chromatic number $\chi_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $\chi_s$ of $d$-complexes can become arbitrarily large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $\chi_{d+1}=1$, and only little is known on $\chi_s$ for $\lceil d/2\rceil<s\leq d$. A particular class of $d$-complexes are triangulations of $d$-manifolds. As a consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that $\chi_2$ for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple systems and is non-constructive. In particular, up to now, no explicit triangulations of surfaces with high $\chi_2$ were known. We show that orientable surfaces of genus at least 20 and non-orientable surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a projective Steiner triple systems, we construct an explicit triangulation of a non-orientable surface of genus 2542 and with face vector $f=(127,8001,5334)$ that has 2-chromatic number 5 or 6. We also give orientable examples with 2-chromatic numbers 5 and 6. For 3-dimensional manifolds, an iterated moment curve construction [18] along with embedding results [6] can be used to produce triangulations with arbitrarily large 2-chromatic number, but of tremendous size. Via a topological version of the geometric construction of [18], we obtain a rather small triangulation of the 3-dimensional sphere $S^3$ with face vector $f=(167,1579,2824,1412)$ and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio

Coloring decompositions of complete geometric graphs

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 $\chi'([G,P])$ 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\H{o}s-Faber-Lov\'asz states that every decomposition $[K_n,P]$ of the complete graph $K_n$ satisfies $\chi'([K_n,P])\leq n$. 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 general position. We also consider the particular case in which the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure

Ramsey theory on Steiner triples

We call a partial Steiner triple system C (configuration) t-Ramsey if for large enough n (in terms of (Formula presented.)), in every t-coloring of the blocks of any Steiner triple system STS(n) there is a monochromatic copy of C. We prove that configuration C is t-Ramsey for every t in three cases: C is acyclic every block of C has a point of degree one C has a triangle with blocks 123, 345, 561 with some further blocks attached at points 1 and 4 This implies that we can decide for all but one configurations with at most four blocks whether they are t-Ramsey. The one in doubt is the sail with blocks 123, 345, 561, 147. © 2017 Wiley Periodicals, Inc

A visual representation of the Steiner triple systems of order 13

Steiner triple systems (STSs) are a basic topic in combinatorics. In an STS the elements can be collected in threes in such a way that any pair of elements is contained in a unique triple. The two smallest nontrivial STSs, with 7 and 9 elements, arise in the context of finite geometry and nonsingular cubic curves, and have well-known pictorial representations. On the contrary, an STS with 13 elements does not have an intrinsic geometric nature, nor a natural pictorial illustration. In this paper we present a visual representation of the two non-isomorphic Steiner triple systems of order 13 by means of a regular hexagram. The thirteen points of each system are the vertices of the twelve equilateral triangles inscribed in the hexagram. In the case of the non-cyclic system, our representation allows one to visualize in a simple, elegant and highly symmetric way the twenty-six triples, the six automorphisms and their orbits, the eight quadrilaterals, the ten mitres, the thirteen grids, the four 3-colouring patterns, the block-colouring and some distinguished ovals. Our construction is based on a very simple idea (seeing the blocks as much as possible as equilateral triangles), which can be further extended to get new representations of the STSs of order 7 and 9, and of one of the STSs of order 15