Coloring and The Lonely Graph

TL;DR

This paper improves upper bounds on the chromatic number of graphs using new lemmas and conditions, generalizes these results to constrained colorings, and applies the findings to longstanding conjectures in graph theory.

Contribution

It introduces a new lemma for paths in graphs, improves bounds on chromatic number, and generalizes Reed's conjecture to r-bounded colorings with applications.

Findings

Improved bounds on the chromatic number based on graph parameters.

Generalization of Reed's conjecture to r-bounded colorings.

Application of results to the Borodin-Kostochka conjecture and graphs with doubly critical edges.

Abstract

We improve upper bounds on the chromatic number proven independently in \cite{reedNote} and \cite{ingo}. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an optimal coloring with more than $\frac{\omega}{2}$ singleton color classes, then it satisfies $\chi \leq \frac{\omega + \Delta + 1}{2}$. It follows that a graph satisfying $n - \Delta < \alpha + \frac{\omega - 1}{2}$ must also satisfy $\chi \leq \frac{\omega + \Delta + 1}{2}$, improving the bounds in \cite{reedNote} and \cite{ingo}. We then give a simple argument showing that if a graph satisfies $\chi > \frac{n + 3 - \alpha}{2}$, then it also satisfies $\chi(G) \leq \left\lceil\frac{\omega(G) + \Delta(G) + 1}{2}\right\rceil$. From this it follows that a graph satisfying $n - \Delta < \alpha + \omega$ also satisfies $\chi(G) \leq \left\lceil\frac{\omega(G) + \Delta(G) + 1}{2}\right\rceil$ improving the bounds in \cite{reedNote} and \cite{ingo} even further at the cost of a ceiling. In the next sections, we generalize our main lemma to constrained colorings (e.g. r-bounded colorings). We present a generalization of Reed's conjecture to r-bounded colorings and prove the conjecture for graphs with maximal degree close to their order. Finally, we outline some applications (in \cite{BorodinKostochka} and \cite{ColoringWithDoublyCriticalEdge}) of the theory presented here to the Borodin-Kostochka conjecture and coloring graphs containing a doubly critical edge.