Destroying Non-Complete Regular Components in Graph Partitions

TL;DR

This paper proves a graph partitioning theorem that ensures each part avoids non-complete regular components, generalizing previous results and providing a method to partition graphs into structures like triangles and paths.

Contribution

It introduces a new partitioning result that prevents non-complete regular components in each part, extending prior work on graph decompositions.

Findings

Partitioning graphs into sets with bounded regular components

Ensures each part contains only complete regular components or specific structures

Applicable to graphs with given maximum degree and sum conditions

Abstract

We prove that if $G$ is a graph and $r_1, ..., r_k \in \mathbb{Z}_{\geq 0}$ such that $\sum_{i=1}^k r_i \geq \Delta(G) + 2 - k$ then $V(G)$ can be partitioned into sets $V_1, ..., V_k$ such that $\Delta(G[V_i]) \leq r_i$ and $G[V_i]$ contains no non-complete $r_i$-regular components for each $1 \leq i \leq k$. In particular, the vertex set of any graph $G$ can be partitioned into $\left \lceil \frac{\Delta(G) + 2}{3} \right \rceil$ sets, each of which induces a disjoint union of triangles and paths.