On the dynamic width of the 3-colorability problem

TL;DR

This paper investigates the minimum logical width needed to distinguish 3-colorability of graphs, establishing bounds related to graph properties and proving that this width grows unboundedly with graph size.

Contribution

It introduces the concept of dynamic width for 3-colorability, providing new bounds and showing its growth rate under various complexity assumptions.

Findings

W(n) is unbounded assuming NP≠P

W(n) has a lower bound of Ω(n) under ETH

W(n) is Θ(√n) for planar graphs

Abstract

A graph $G$ is 3-colorable if and only if it maps homomorphically to the complete 3-vertex graph $K_3$. The last condition can be checked by a $k$-consistency algorithm where the parameter $k$ has to be chosen large enough, dependent on $G$. Let $W(G)$ denote the minimum $k$ sufficient for this purpose. For a non-3-colorable graph $G$, $W(G)$ is equal to the minimum $k$ such that $G$ can be distinguished from $K_3$ in the $k$-variable existential-positive first-order logic. We define the dynamic width of the 3-colorability problem as the function $W(n)=\max_G W(G)$, where the maximum is taken over all non-3-colorable $G$ with $n$ vertices. The assumption $\mathrm{NP}\ne\mathrm{P}$ implies that $W(n)$ is unbounded. Indeed, a lower bound $W(n)=\Omega(\log\log n/\log\log\log n)$ follows unconditionally from the work of Nesetril and Zhu on bounded treewidth duality. The Exponential Time Hypothesis implies a much stronger bound $W(n)=\Omega(n/\log n)$ and indeed we unconditionally prove that $W(n)=\Omega(n)$. In fact, an even stronger statement is true: A first-order sentence distinguishing any 3-colorable graph on $n$ vertices from any non-3-colorable graph on $n$ vertices must have $\Omega(n)$ variables. On the other hand, we observe that $W(G)\le 3\,\alpha(G)+1$ and $W(G)\le n-\alpha(G)+1$ for every non-3-colorable graph $G$ with $n$ vertices, where $\alpha(G)$ denotes the independence number of $G$. This implies that $W(n)\le\frac34\,n+1$, improving on the trivial upper bound $W(n)\le n$. We also show that $W(G)>\frac1{16}\, g(G)$ for every non-3-colorable graph $G$, where $g(G)$ denotes the girth of $G$. Finally, we consider the function $W(n)$ over planar graphs and prove that $W(n)=\Theta(\sqrt n)$ in the case.