Minimum degree and sparse connected spanning subgraphs

TL;DR

This paper proves that a graph with certain degree and edge constraints contains a given sparse graph as a spanning subgraph, extending classical results on trees to more general graphs using tight bounds on Ramsey numbers.

Contribution

It generalizes previous results by establishing spanning subgraph conditions for broader classes of graphs using new bounds on Ramsey numbers.

Findings

Proves $F$ contains $G$ as a spanning subgraph under specified conditions.

Establishes tight bounds for the Ramsey number $r(G,K_{1,k})$.

Extends classical results from trees to more general sparse graphs.

Abstract

Let $G$ be a connected graph on $n$ vertices and at most $n(1+\epsilon)$ edges with bounded maximum degree, and $F$ a graph on $n$ vertices with minimum degree at least $n-k$, where $\epsilon$ is a constant depending on $k$. In this paper, we prove that $F$ contains $G$ as a spanning subgraph provided $n\ge 6k^3$, by establishing tight bounds for the Ramsey number $r(G,K_{1,k})$, where $K_{1,k}$ is a star on $k+1$ vertices. Our result generalizes and refines the work of Erd\H{o}s, Faudree, Rousseau, and Schelp (JCT-B, 1982), who established the corresponding result for $G$ being a tree. Moreover, the tight bound for $r(G,tK_{1,k})$ is also obtained.