Universal Sets for Straight-Line Embeddings of Bicolored Graphs

TL;DR

This paper investigates conditions under which certain point sets, specifically double chains, can universally embed bicolored graphs with various coloring constraints, revealing limitations and capabilities for straight-line embeddings.

Contribution

It introduces new bounds and conditions for the universality of double chains in embedding equitable and bipartite graphs, expanding understanding of geometric graph embeddings.

Findings

Double chains are 2-color universal for paths with balanced chain sizes.

Universality fails if one chain is significantly longer than the other.

Equitable caterpillars with limited non-leaf vertices can be embedded on certain double chains.

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 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.