An Ore-type theorem for perfect packings in graphs

TL;DR

This paper establishes asymptotic Ore-type degree conditions that guarantee the existence of perfect H-packings in large graphs, extending classical packing theorems to a broader degree condition framework.

Contribution

It determines the asymptotic Ore-type degree threshold for perfect H-packings in graphs, generalizing previous degree conditions for graph packings.

Findings

Calculated the limit of the Ore-type degree threshold divided by n for large graphs.

Provided a general asymptotic condition for perfect packings based on Ore-type degree sum.

Extended classical packing results to Ore-type degree conditions.

Abstract

We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let \delta_{\rm Ore} (H,n) be the smallest number k such that every graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine \lim_{n\to \infty} \delta_{\rm Ore} (H,n)/n.