A Superlocal Version of Reed's Conjecture

Katherine Edwards,Andrew D King

doi:10.37236/2666

Katherine Edwards, Andrew D King

Open Access

https://doi.org/10.37236/2666

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Abstract

Reed's well-known $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisfies $\chi \leq \lceil \frac 12(\Delta+1+\omega)\rceil$. The second author formulated a local strengthening of this conjecture that considers a bound supplied by the neighbourhood of a single vertex. Following the idea that the chromatic number cannot be greatly affected by any particular stable set of vertices, we propose a further strengthening that considers a bound supplied by the neighbourhoods of two adjacent vertices. We provide some fundamental evidence in support, namely that the stronger bound holds in the fractional relaxation and holds for both quasi-line graphs and graphs with stability number two. We also conjecture that in the fractional version, we can push the locality even further.

Highlights

We consider simple graphs with clique number ω, maximum degree ∆, chromatic number χ, and fractional chromatic number χf
Brooks’ Theorem states that whenever ∆ 3, a graph with maximum degree ∆ is ∆-colourable unless it has the obvious obstruction: a clique of size ∆ + 1
A graph with maximum degree ∆ is (∆ + 1 − k)-colourable unless it contains a clique of size at least ∆ + 2 − 2k

Summary

Introduction

We consider simple graphs with clique number ω, maximum degree ∆, chromatic number χ, and fractional chromatic number χf (we will define χf later). A graph with maximum degree ∆ is (∆ + 1 − k)-colourable unless it contains a clique of size at least ∆ + 2 − 2k. This conjecture is known to hold for claw-free graphs [10] and some other hereditary families of graphs [1]. A typical example of a graph G for which γ(G) is far from χ(G) is the star K1,r For such graphs we have γ (G) = γ(G), so the bound offered by the local conjecture isn’t any better. Can we get a better bound when vertices that are hard to colour (i.e. have high γ (v)) form a stable set? The answer, at least in the fractional setting and for certain graph classes, is yes

The superlocal strengthening

Proving the fractional relaxation

Some easy integer colouring cases

Conclusion