Triangle-tilings in graphs without large independent sets

TL;DR

This paper determines the minimum degree conditions needed for perfect and almost-perfect triangle-tilings in graphs with small independence number, advancing understanding of tiling thresholds under various constraints.

Contribution

It establishes new minimum degree thresholds for triangle-tilings in graphs with sublinear independence number, including conditions for perfect tilings in $K_r$-free graphs.

Findings

Graphs with minimum degree at least n/3 + o(n) have almost-perfect triangle-tilings.

Exact thresholds are identified for perfect triangle-tilings in $K_r$-free graphs for $r \,\geq 5$.

Results extend tiling theory to graphs with small independence number and forbidden subgraphs.

Abstract

We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an $n$-vertex graph $G$ with sublinear independence number. In this setting, we show that if $\delta(G) \ge n/3 + o(n)$ then $G$ has a triangle-tiling covering all but at most four vertices. Also, for every $r \ge 5$, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that $G$ is $K_r$-free and $n$ is divisible by $3$.