
TL;DR
This paper investigates the property of graphs called linear freeness, which depends on their embeddings in three-dimensional space, and establishes conditions under which graphs maintain this property, with applications to complete and bipartite graphs.
Contribution
It generalizes Nicholson's concept of linear freeness, providing new criteria for when a graph's linear embedding is free and exploring how this property is preserved under graph enlargements.
Findings
Complete graphs are linearly free.
Graphs with minimal valency ≥ 3 and fewer than 8 vertices are linearly free.
Complete bipartite graphs K_{n,m} are linearly free for n,m ≤ 6.
Abstract
In this paper we are interested in an intrinsic property of graphs which is derived from their embeddings into the Euclidean 3-space . An embedding of a graph into is said to be linear, if it sends every edge to be a line segment. And we say that an embedding of a graph into is free, if is a free group. Lastly a simple connected graph is said to be linearly free if every its linear embedding is free. In 1980s it was proved that every complete graph is linearly free, by Nicholson. In this paper, we develop Nicholson's arguments into a general notion, and establish a sufficient condition for a linear embedding to be free. As an application of the condition we give a partial answer for a question: how much can the complete graph be enlarged so that the linear freeness is preserved and the clique number…
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Taxonomy
TopicsFinite Group Theory Research · Geometric and Algebraic Topology · Structural Analysis and Optimization
