On the decomposition threshold of a given graph

TL;DR

This paper investigates the minimum degree threshold needed for a large graph to be decomposed into copies of a fixed smaller graph, providing bounds and exact values for various classes of graphs.

Contribution

The authors establish new bounds and exact thresholds for the $F$-decomposition problem, extending the iterative absorbing method to this context.

Findings

Bound $oldsymbol{ ext{delta}_F extless= ext{max}igrace{ ext{delta}_F^*, 1 - 1/( ext{chi}+1)}igrace}$ for the decomposition threshold.

Determine $oldsymbol{ ext{delta}_F}$ exactly for bipartite graphs.

Identify possible values of $oldsymbol{ ext{delta}_F}$ when $ ext{chi} extgreater= 5$.

Abstract

We study the $F$-decomposition threshold $\delta_F$ for a given graph $F$. Here an $F$-decomposition of a graph $G$ is a collection of edge-disjoint copies of $F$ in $G$ which together cover every edge of $G$. (Such an $F$-decomposition can only exist if $G$ is $F$-divisible, i.e. if $e(F)\mid e(G)$ and each vertex degree of $G$ can be expressed as a linear combination of the vertex degrees of $F$.) The $F$-decomposition threshold $\delta_F$ is the smallest value ensuring that an $F$-divisible graph $G$ on $n$ vertices with $\delta(G)\ge(\delta_F+o(1))n$ has an $F$-decomposition. Our main results imply the following for a given graph $F$, where $\delta_F^\ast$ is the fractional version of $\delta_F$ and $\chi:=\chi(F)$: (i) $\delta_F\le \max\{\delta_F^\ast,1-1/(\chi+1)\}$; (ii) if $\chi\ge 5$, then $\delta_F\in\{\delta_F^{\ast},1-1/\chi,1-1/(\chi+1)\}$; (iii) we determine $\delta_F$ if $F$ is bipartite. In particular, (i) implies that $\delta_{K_r}=\delta^\ast_{K_r}$. Our proof involves further developments of the recent `iterative' absorbing approach.