Weak diameter choosability of graphs with an excluded minor
Joshua Crouch, Chun-Hung Liu
TL;DR
This work establishes that every $H$-minor free graph is $3$-choosable with a bounded weak diameter, with the constant depending on $H$; the bound is tight for non-planar $H$, while planar exclusions admit a stronger planar-special extension that leverages a precoloring of bounded weak diameter. The approach integrates Robertson–Seymour structure theory, near-embeddings in bounded-genus surfaces, and a new list-coloring extension framework built around legitimate and extendable list-assignments and $( ext{η}, ext{θ})$-constructions. A key advance is showing that precolorings with bounded weak diameter can be extended to full list-colorings with bounded weak diameter in $H$-minor free graphs when the excluded minor is planar, yielding exponentially many colorings and clustering-controlled colorings for related graph families. The paper also identifies limits in the odd-minor regime, proving lower bounds for bipartite graphs on the required list-size to achieve bounded weak diameter, while proving 3-colorability with bounded weak diameter for odd $H$-minor free graphs in the non-list setting. Collectively, these results deepen the understanding of how structural graph decompositions interact with weak diameter constraints to produce robust, copyable colorings in minor-closed classes, with implications for clustering, layering, and geometric graph families.
Abstract
Weak diameter coloring of graphs recently attracted attention partially due to its connection to asymptotic dimension of metric spaces. We consider weak diameter list-coloring of graphs in this paper. Dvořák and Norin proved that graphs with bounded Euler genus are 3-choosable with bounded weak diameter. In this paper, we extend their result by showing that for every graph $H$, $H$-minor free graphs are 3-choosable with bounded weak diameter. The upper bound 3 is optimal and it strengthens an earlier result for non-list-coloring $H$-minor free graphs with bounded weak diameter. As a corollary, $H$-minor free graphs with bounded maximum degree are 3-choosable with bounded clustering, strengthening an earlier result for non-list-coloring. When $H$ is planar, we prove a much stronger result: for every 2-list-assignment $L$ of an $H$-minor free graph, every precoloring with bounded weak diameter can be extended to an $L$-coloring with bounded weak diameter. As a corollary, for any planar graph $H$ and $H$-minor free graph $G$, there are exponentially many list-colorings of $G$ with bounded weak diameter (and with bounded clustering if $G$ also has bounded maximum degree); and every graph with bounded layered tree-width and bounded maximum degree has exponentially many 3-colorings with bounded clustering. We also show that the aforementioned results for list-coloring cannot be extended to odd minor free graphs by showing that some bipartite graphs with maximum degree $Δ$ are $k$-choosable with bounded weak diameter only when $k=Ω(\logΔ/\log\logΔ)$. On the other hand, we show that odd $H$-minor graphs are 3-colorable with bounded weak diameter, implying an earlier result about clustered coloring of odd $H$-minor free graphs with bounded maximum degree.