Odd wheels are not odd-distance graphs

TL;DR

This paper proves that odd wheel graphs cannot be embedded in the plane with all edges having odd integer lengths, addressing a specific question about graph embeddings and edge length constraints.

Contribution

The paper establishes a new geometric property of odd wheel graphs, demonstrating their incompatibility with odd-integer edge length planar embeddings.

Findings

Odd wheel graphs cannot be embedded with all edges of odd integer length.

Addresses a previously open question by Rosenfeld and Le.

Provides insights into geometric constraints of specific graph classes.

Abstract

An odd wheel graph is a graph formed by connecting a new vertex to all vertices of an odd cycle. We answer a question of Rosenfeld and Le by showing that odd wheels cannot be drawn in the plane such that the lengths of the edges are odd integers.