The Tur\'{a}n Number for Spanning Linear Forests

TL;DR

This paper determines the maximum number of edges in large graphs that do not contain certain spanning linear forests, extending Turán-type extremal graph theory results.

Contribution

It establishes an asymptotic formula for the extremal number for spanning linear forests with many edges, generalizing classical Turán problems.

Findings

Derived an asymptotic formula for ex(n;L_n^k)

Applicable when n ≥ 3k and k ≥ 2

Result is significant for k=o(n)

Abstract

For a set of graphs $\mathcal{F}$, the extremal number $ex(n;\mathcal{F})$ is the maximum number of edges in a graph of order $n$ not containing any subgraph isomorphic to some graph in $\mathcal{F}$. If $\mathcal{F}$ contains a graph on $n$ vertices, then we often call the problem a spanning Tur\'{a}n problem. A linear forest is a graph whose connected components are all paths and isolated vertices. In this paper, we let $\mathcal{L}_n^k$ be the set of all linear forests of order $n$ with at least $n-k+1$ edges. We prove that when $n\geq 3k$ and $k\geq 2$, \[ ex(n;\mathcal{L}_n^k)=\binom{n-k+1}{2}+ O(k^2). \] Clearly, the result is interesting when $k=o(n)$.