Universal Sets for Straight-Line Embeddings of Bicolored Graphs

Abstract

A set S of n points is 2-color universal for a graph G on n vertices if for every proper 2-coloring of G and for every 2-coloring of S with * Research was supported by the project 1M0545 of the Ministry of Education of the Czech Republic and by the grant SVV-2010-261313 (Discrete Methods and Algorithms). Viola Mészáros was also partially supported by OTKA grant K76099 and by European project IST-FET AEOLUS. Josef Cibulka and Rudolf Stolař were also supported by the Czech Science Foundation under the contract no. 201/09/H057. Josef Cibulka, Jan Kynčl and Pavel Valtr were also supported by the Grant Agency of the Charles University, GAUK 52410. Part of the research was conducted during the Special Semester on Discrete and Computational Geometry atÉcole Polytechnique Fédérale de Lausanne, organized and supported by the CIB (Centre Interfacultaire Bernoulli) and the SNSF (Swiss National Science Foundation).† A preliminary version appeared in proceedings of Graph Drawing 2008 [7].1 the same sizes of color classes as G has, G is straight-line embeddable on S.We show that the so-called double chain is 2-color universal for paths if each of the two chains contains at least one fifth of all the points, but not if one of the chains is more than approximately 28 times longer than the other.A 2-coloring of G is equitable if the sizes of the color classes differ by at most 1. A bipartite graph is equitable if it admits an equitable proper coloring. We study the case when S is the double-chain with chain sizes differing by at most 1 and G is an equitable bipartite graph. We prove that this S is not 2-color universal if G is not a forest of caterpillars and that it is 2-color universal for equitable caterpillars with at most one half non-leaf vertices. We also show that if this S is equitably 2-colored, then equitably properly 2-colored forests of stars can be embedded on it.

On Hamiltonian alternating cycles and paths

“…However, the conjecture that | (n) − 4 3 n| = o(n) remains open, even for points in convex position. For more information on this topic see [2,5].…”

“…Other references on bicolored point sets are [13] (which is the book of reference) and [7,8,20] (where monochromatic paths and matchings are studied). See also [5] for results on alternating paths on double chains, and [2] for results on plane alternating spanning tress. For other problems on geometric graphs see [4,12,18].…”

Long Alternating Paths Exist

“…All points are covered by the chunks, and there are n red points and n blue points. This implies (5). Since all points in P are covered by the chunks in Γ, the lower bound R + B ≥ 2n/(2k − 1) from (3) becomes R + B ≥ 2n/(2k − 1) ≥ n/k, and hence also max{R, B} ≥ n/2k.…”

“…R + B) C red,D red k + C red,D blue max{b(C), r(D)} + C blue,D red max{r(C), b(D)} + C blue,D blue k .There are R 2 pairs of red chunks and B 2 pairs of blue chunks, and the maximum can be lower bounded by the average, so this is≥ 1 2(R + B) kR 2 + C red,D blue b(C) + r(D) 2 + C blue,D red r(C) + b(D) 2 +Since there are αkR blue points in the red chunks, and βbB red points in the blue chunks, this is= kR 2 + αkBR + βkBR + kB 2 2(R + B)Regrouping the terms and using(5), this equals = R(kR + βkB) + B(kB + αkR)…”

The Chromatic Number of the Convex Segment Disjointness Graph

“…The double chain was first introduced by Hurtado, Noy and Urrutia in [11] as an example of a set of n points (in general position) whose flip graph of triangulations has diameter Θ(n 2 ). Since then the double chain has been used as an extremal example in various problems on point sets, see for example [1,2,3,5,6,7,10].…”