Fractional and $j$-fold colouring of the plane

TL;DR

This paper explores fractional and $j$-fold colourings of the plane, providing bounds and methods for graphs with edges in specific distance intervals, with implications for the Hadwiger-Nelson problem and applications in scheduling and radio networks.

Contribution

It generalizes fractional colouring bounds for graphs $G_{[a,b]}$, offers new methods for $j$-fold colourings, and partially addresses a conjecture related to the Hadwiger-Nelson problem.

Findings

Proved $ ext{chi}(G_{[a,b]}) ext{geq} 5$ for $b > a$.

Extended fractional colouring bounds depending on $rac{a}{b}$.

Developed specific and general methods for small $j$-fold colourings of $G_{[a,b]}$.

Abstract

We present results referring to the Hadwiger-Nelson problem which asks for the minimum number of colours needed to colour the plane with no two points at distance $1$ having the same colour. Exoo considered a more general problem concerning graphs $G_{[a,b]}$ with $\mathbb{R}^2$ as the vertex set and two vertices adjacent if their distance is in the interval $[a,b]$. Exoo conjectured $\chi(G_{[a,b]}) = 7$ for sufficiently small but positive difference between $a$ and $b$. We partially answer this conjecture by proving that $\chi(G_{[a,b]}) \geq 5$ for $b > a$. A $j$-fold colouring of graph $G = (V,E)$ is an assignment of $j$-elemental sets of colours to the vertices of $G$, in such a way that the sets assigned to any two adjacent vertices are disjoint. The fractional chromatic number $\chi_f(G)$ is the infimum of fractions $k/j$ for $j$-fold colouring of $G$ using $k$ colours. We generalize a method by Hochberg and O'Donnel (who proved that $G_{[1,1]} \leq 4.36$) for fractional colouring of graphs $G_{[a,b]}$, obtaining a bound dependant on $\frac{a}{b}$. We also present few specific and two general methods for $j$-fold coloring of $G_{[a,b]}$ for small $j$, in particular for $G_{[1,1]}$ and $G_{[1,2]}$. The $j$-fold colouring for small $j$ has strong practical motivation especially in scheduling theory, while graph $G_{[1,2]}$ is often used to model hidden conflicts in radio networks.