TSTST 2026 Problem 2

Given a graph $G$ and a positive integer $\ell$, an $\ell$-coloring of $G$ is a function $\sigma \colon V(G) \to \{1, 2, 3, \ldots, \ell\}$ such that for any edge $uv$ in $G$, $\sigma(u) \neq \sigma(v)$. Two $\ell$-colorings $\sigma$ and $\sigma'$ are close if $\sigma(u) \neq \sigma'(u)$ for at most one $u \in V(G)$.

Find all pairs of positive integers $(k, m)$ with $k \leq m$ such that for all graphs $G$ and pairs $(\tau, \tau')$ of $k$-colorings of $G$, there exists a sequence $\pi_0, \pi_1, \ldots, \pi_n$ of $m$-colorings of $G$ such that $\pi_0 = \tau$, $\pi_n = \tau'$, and $\pi_i$ and $\pi_{i+1}$ are close for all $0 \leq i < n$.

Note: $V(G)$ denotes the vertices of $G$. Since $k \leq m$, $\tau$ and $\tau'$ are also $m$-colorings of $G$.