Guessing Numbers of Odd Cycles

TL;DR

This paper determines the guessing number of odd cycle graphs with any number of colours, providing an explicit formula and protocol, and linking it to index coding with side information.

Contribution

It generalizes the guessing number for odd cycles with arbitrary colours, solving a question from 2011 and providing explicit protocols.

Findings

Guessing number for large odd cycles with s colours is (n-1)/2 + log_s(a).

An explicit protocol achieves this bound for all n.

The information defect is (n+1)/2 - log_s(a) for large odd n.

Abstract

For a given number of colours, $s$, the guessing number of a graph is the base $s$ logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the $n$-vertex cycle graph $C_n$ is $n/2$. It is known that the guessing number equals $n/2$ whenever $n$ is even or $s$ is a perfect square \cite{Christofides2011guessing}. We show that, for any given integer $s\geq 2$, if $a$ is the largest factor of $s$ less than or equal to $\sqrt{s}$, for sufficiently large odd $n$, the guessing number of $C_n$ with $s$ colours is $(n-1)/2 + \log_s(a)$. This answers a question posed by Christofides and Markstr\"{o}m in 2011 \cite{Christofides2011guessing}. We also present an explicit protocol which achieves this bound for every $n$. Linking this to index coding with side information, we deduce that the information defect of $C_n$ with $s$ colours is $(n+1)/2 - \log_s(a)$ for sufficiently large odd $n$. Our results are a generalisation of the $s=2$ case which was proven in \cite{bar2011index}.