Saturation Number of Trees in the Hypercube

TL;DR

This paper studies the minimum number of edges in hypercube subgraphs that are saturated with respect to trees, providing bounds and construction methods for various types of trees.

Contribution

It introduces new bounds and construction techniques for the saturation number of trees in hypercubes, advancing understanding of hypercube subgraph extremal properties.

Findings

Established a lower bound based on non-leaf degree

Proposed two methods for constructing T-saturated subgraphs

Derived upper bounds for paths, stars, and caterpillars

Abstract

A graph $H^{\prime}$ is $(H, G)$-saturated if it is $G$-free and the addition of any edge of $H$ not in $H^{\prime}$ creates a copy of $G$. The saturation number $sat(H, G)$ is the minimum number of edges in a $(H, G)$-saturated graph. We investigate bounds on the saturation number of trees $T$ in the $n$-dimensional hypercube $Q_n$. We first present a general lower bound on the saturation number based on the minimum degree of non-leaves. From there, we suggest two general methods for constructing $T$-saturated subgraphs of $Q_n$, and prove nontrivial upper bounds for specific types of trees, including paths, generalized stars, and certain caterpillars under a restriction on minimum degree with respect to diameter.