On star-forest ascending subgraph decomposition

TL;DR

This paper proves that certain bipartite graphs with specific degree sequences can be decomposed into ascending star-forest subgraphs, advancing understanding of the ASD Conjecture.

Contribution

It establishes a sufficient condition for bipartite graphs to admit an ascending subgraph decomposition into star forests, and provides a near-necessary degree sequence condition.

Findings

Bipartite graphs with specified degree sequences admit star-forest ASD.

A sufficient degree sequence condition for ASD in bipartite graphs.

A near-necessary condition on degree sequences for ASD.

Abstract

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph $G$ with ${n+1\choose 2}$ edges admits an edge decomposition $G=H_1\oplus\cdots \oplus H_n$ such that $H_i$ has $i$ edges and it is isomorphic to a subgraph of $H_{i+1}$, $i=1,\ldots ,n-1$. We show that every bipartite graph $G$ with ${n+1\choose 2}$ edges such that the degree sequence $d_1,\ldots ,d_k$ of one of the stable sets satisfies $ d_{k-i}\ge n-i\; \text{for each}\; 0\le i\le k-1,$, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.