A Proof of Delta Conjecture

TL;DR

This paper proves the delta conjecture by constructing orthogonal representations for specific classes of graphs, extending the approach to more complex graph structures, and establishing the conjecture's validity.

Contribution

It introduces a method to prove the delta conjecture using orthogonal representations for $ ext{delta}$-graphs and their extensions, providing a new proof for the conjecture.

Findings

Delta graphs satisfy the delta conjecture.

Orthogonal representations exist for extended graph classes.

The proof applies to graphs of the form $ar{P_{ ext{delta}+2} old G}$.

Abstract

By finding orthogonal representation for a family of simple connected called $\delta$-graphs it is possible to show that $\delta$-graphs satisfy delta conjecture. An extension of the argument to graphs of the form $\overline{P_{\Delta(G)+2}\sqcup G}$ where $P_{\Delta(G)+2}$ is a path and $G$ is a simple connected graph it is possible to find an orthogonal representation of $\overline{P_{\Delta(G)+2}\sqcup G}$ in $\mathbb{R}^{\Delta(G)+1}$. As a consequence we prove delta conjecture.