Partitioning $H$-minor free graphs into three subgraphs with no large components

TL;DR

This paper proves that graphs excluding a fixed minor can be partitioned into three parts with small components, improving previous bounds and extending results from surface-embeddable graphs to all minor-free graphs.

Contribution

It establishes a new partitioning theorem for minor-free graphs, reducing the number of parts needed compared to earlier results, and generalizes prior surface-embeddable graph findings.

Findings

Graphs with no (odd) H minor can be partitioned into three parts with bounded component size.

Improves the partition count from four to three for H-minor free graphs.

Provides a partitioning bound for graphs excluding a complete graph minor.

Abstract

We prove that for every graph $H$, if a graph $G$ has no (odd) $H$ minor, then its vertex set $V(G)$ can be partitioned into three sets $X_1$, $X_2$, $X_3$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$H$ and the maximum degree of~$G$. This improves a previous result of Alon, Ding, Oporowski and Vertigan~(2003) stating that $V(G)$ can be partitioned into four such sets if $G$ has no $H$ minor. Our theorem generalizes a result of Esperet and Joret~(2014), who proved it for graphs embeddable on a fixed surface and asked whether it is true for graphs with no $H$ minor. As a corollary, we prove that for every positive integer $t$, if a graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $3t$ sets $X_1,\ldots,X_{3t}$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$t$. This corollary improves a result of Wood~(2010), which states that $V(G)$ can be partitioned into $\lceil 3.5t+2\rceil$ such sets.