Ascending Subgraph Decomposition

TL;DR

This paper proves a conjecture from 1987 that large graphs with a specific number of edges can be decomposed into a sequence of subgraphs with increasing size, each isomorphic to a subgraph of the next.

Contribution

It confirms the conjecture for sufficiently large values of m, establishing a new result in graph decomposition theory.

Findings

Proves the conjecture for large m

Shows existence of ascending subgraph decompositions

Advances understanding of graph decomposition conditions

Abstract

A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph $G$ into copies $H_1, \ldots, H_m$ are also sufficient. One such problem was posed in 1987, by Alavi, Boals, Chartrand, Erd\H{o}s, and Oellerman. They conjectured that the edges of every graph with $\binom{m+1}2$ edges can be decomposed into subgraphs $H_1, \dots, H_m$ such that each $H_i$ has $i$ edges and is isomorphic to a subgraph of $H_{i+1}$. In this paper we prove this conjecture for sufficiently large $m$.