On the range of two-distance graphs

Abstract

The topic of this paper is related to the well-known notion of unit distance graphs. Take a graph with its edges coloured red and blue such that for some $d$ it can be mapped into the plane with all vertices going to distinct points, the red edges to segments of length $1$ and the blue edges to segments of length $d$. We define the range of this graph to be the set of such numbers $d$. It is easy to show that the range of any edge-bicoloured graph consists of finitely many intervals with algebraic endpoints, and we now prove that any such set with a finite positive upper and lower bound is the range of a suitable graph.

Figures (12)

• Figure 1: A polynomial $p(x)$ with $S_0\left(p,L,U\right)$ (left) and $S_1\left(p,L,U\right)$ (right) denoted by bold
• Figure 2: The components of $A$: $A_1$ (left), the only $(1,d)$-representations (up to isometry) of $A_1$ (middle) and of $A_j$ ($2\le j\le deg(p)$) (right) ($N$ is large enough and groups are denoted by numbers).
• Figure 3:
• Figure 4: Graph $A_1$. The dashed edge only represents that typically there there is more than one edge between $a_{1,6}$ and $a_{1,2N-1}$
• Figure 5: The only possible $(1,d)$-representation of $A_1$ up to isometry for some $d$

Theorems & Definitions (63)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof