Isolation partitions in graphs

TL;DR

This paper introduces new partitioning results for graphs, showing how certain graphs can be divided into disjoint sets that isolate specific substructures like cliques and cycles, with implications for regular graphs.

Contribution

It proves that connected graphs with bounded degree can be partitioned into disjoint isolating sets for cliques and cycles, extending to regular graphs and claw-free graphs.

Findings

Connected graphs with max degree ≤ k (except k-clique) can be partitioned into k+1 disjoint k-clique isolating sets.

Connected claw-free subcubic graphs (except 3-cycle) can be partitioned into four disjoint cycle isolating sets.

Every k-regular graph can be partitioned into k+1 disjoint k-clique isolating sets.

Abstract

Let $G$ be a graph and $k \geq 3$ an integer. A subset $D \subseteq V(G)$ is a $k$-clique (resp., cycle) isolating set of $G$ if $G-N[D]$ contains no $k$-clique (resp., cycle). In this paper, we prove that every connected graph with maximum degree at most $k$, except $k$-clique, can be partitioned into $k+1$ disjoint $k$-clique isolating sets, and that every connected claw-free subcubic graph, except 3-cycle, can be partitioned into four disjoint cycle isolating sets. As a consequence of the first result, every $k$-regular graph can be partitioned into $k+1$ disjoint $k$-clique isolating sets.