List Colouring Trees in Logarithmic Space

TL;DR

This paper demonstrates that List Colouring on trees can be efficiently solved in logarithmic space, significantly improving the understanding of space complexity for graph colouring problems.

Contribution

It introduces a deterministic logarithmic space algorithm for List Colouring on trees, advancing the computational complexity knowledge for this problem.

Findings

List Colouring on trees is solvable in O(log n) space.

The algorithm is deterministic and space-efficient.

This work improves previous complexity bounds for tree colourings.

Abstract

We show that List Colouring can be solved on $n$-vertex trees by a deterministic Turing machine using $O(\log n)$ bits on the worktape. Given an $n$-vertex graph $G=(V,E)$ and a list $L(v)\subseteq\{1,\dots,n\}$ of available colours for each $v\in V$, a list colouring for $G$ is a proper colouring $c$ such that $c(v)\in L(v)$ for all $v$.