On Orthogonal Vector Edge Coloring

TL;DR

This paper explores the edge version of orthogonal vector coloring in graphs, examining its relation to known graph parameters and posing open questions about the orthogonal chromatic index of cubic graphs.

Contribution

It introduces the edge version of orthogonal vector coloring, analyzes its connections with existing graph invariants, and highlights open problems in the area.

Findings

Discusses the relation between orthogonal vector edge coloring and Lovász Theta Function.

Identifies a knowledge gap in the edge version of the problem.

Poses open questions about the orthogonal chromatic index of cubic graphs.

Abstract

Given a graph $G$ and a positive integer $d$, an orthogonal vector $d$-coloring of $G$ is an assignment $f$ of vectors of $\mathbb{R}^d$ to $V(G)$ in such a way that adjacent vertices receive orthogonal vectors. The orthogonal chromatic number of $G$, denoted by $\chi_v(G)$, is the minimum $d$ for which $G$ admits an orthogonal vector $d$-coloring. This notion has close ties with the notions of Lov\'asz Theta Function, quantum chromatic number, and many other problems, and even though this and related metrics have been extensively studied over the years, we have found that there is a gap in the knowledge concerning the edge version of the problem. In this article, we discuss this version and its relation with other insteresting known facts, and pose a question about the orthogonal chromatic index of cubic graphs.