An improved error term for minimum H-decompositions of graphs

TL;DR

This paper refines the bounds on the minimum number of parts needed to partition a graph's edges into copies of a fixed graph H and single edges, improving previous results by providing a sharper error term for graphs with chromatic number greater than two.

Contribution

It introduces a new error term for minimum H-decompositions, improving existing bounds and extending prior results in graph decomposition theory.

Findings

Established that _H(n) = ex(n,K_r) + \u039b(biex(n,H)) for hi(H)>2.

Showed biex(n,H) = O(n^{2-\u03b3}) for some bb, leading to tighter bounds.

Extended previous results by Pikhurko, Sousa, fOzkahya, and Person.

Abstract

We consider partitions of the edge set of a graph G into copies of a fixed graph H and single edges. Let \phi_H(n) denote the minimum number p such that any n-vertex G admits such a partition with at most p parts. We show that \phi_H(n)=ex(n,K_r)+\Theta(biex(n,H)) for \chi(H)>2, where biex(n,H) is the extremal number of the decomposition family of H. Since biex(n,H)=O(n^{2-\gamma}) for some \gamma>0 this improves on the bound \phi_H(n)=ex(n,H)+o(n^2) by Pikhurko and Sousa [J. Combin. Theory Ser. B 97 (2007), 1041-1055]. In addition it extends a result of \"Ozkahya and Person [J. Combin. Theory Ser. B, to appear].