Quasi-tree-partitions of graphs with an excluded subgraph

TL;DR

This paper introduces quasi-tree-partitions to describe graphs with bounded tree-width excluding a fixed bipartite subgraph, and applies this to improve understanding of fractional clustered coloring.

Contribution

It provides a new structural framework for such graphs and extends results to fractional and clustered coloring applications.

Findings

Structural description of graphs with bounded tree-width excluding a bipartite subgraph

Applications to fractional clustered coloring

Generalizations of structural and coloring results

Abstract

This paper studies the structure of graphs with given tree-width and excluding a fixed complete bipartite subgraph, which generalises the bounded degree setting. We give a new structural description of such graphs in terms of so-called quasi-tree-partitions. We demonstrate the utility of this result through applications to (fractional) clustered colouring. Further generalisations of these structural and colouring results are presented.