A relative of Hadwiger's conjecture

TL;DR

This paper explores a related partitioning property of graphs with no $K_{t+1}$ minor, showing that their vertices can be divided into t parts with bounded degree subgraphs, extending Hadwiger's conjecture.

Contribution

It proves a new partitioning result for graphs with no $K_{t+1}$ minor, where each part induces a subgraph with bounded maximum degree, a significant extension of Hadwiger's conjecture.

Findings

Partition into t sets with degree bounds is sharp.

The result does not hold for t-1 sets.

Supports a relative of Hadwiger's conjecture.

Abstract

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned into $t$ sets $X_1,\ldots, X_t$, such that for $1\le i\le t$, the subgraph induced on $X_i$ has maximum degree at most a function of $t$. This is sharp, in that the conclusion becomes false if we ask for a partition into $t-1$ sets with the same property.