Forbidden subgraphs and complete partitions

TL;DR

This paper investigates the properties of graphs that can be partitioned into parts with limited size and specific edge conditions, providing improved bounds on the minimum size of parts avoiding certain subgraphs, and confirming some conjectures.

Contribution

The paper improves bounds on the function f(r,H) for H being a complete bipartite graph or an even cycle, advancing understanding of forbidden subgraph structures.

Findings

Some bounds are tight up to a constant factor.

Confirmed a conjecture of Axenovich and Martin in several cases.

Provided new insights into the structure of H-free (r,k)-graphs.

Abstract

A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts, each having at most $k$ vertices and there is at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an $H$-free $(r,k)$-graph. In this paper we build on the work of Axenovich and Martin, obtaining improved bounds on this function when $H$ is a complete bipartite graph or an even cycle. Some of these bounds are best possible up to a constant factor and confirm a conjecture of Axenovich and Martin in several cases.