On the Maximum Order of Induced Paths and Induced Forests in Regular Graphs

TL;DR

This paper explores bounds on the maximum size of induced forests and linear forests in regular graphs, extending known results and proposing a conjecture relating degree sequences to induced linear forests.

Contribution

It generalizes existing bounds for induced forests to induced linear forests in regular graphs and proposes a new conjecture for broader classes of graphs.

Findings

Established that LIF(G) ≥ (2/(r+1)) * n for r-regular graphs.

Extended known bounds on induced forests to induced linear forests.

Proposed a conjecture relating degree sequences to the size of induced linear forests.

Abstract

Let $G$ be a graph and $a(G)$, LIF$(G)$ denote the maximum orders of an induced forest and an induced linear forest of $G$, respectively. It is well-known that if $G$ is an $r$-regular graph of order $n$, then $a(G) \geq \frac{2}{r+1}n$. In this paper, we generalize this result by showing that LIF$(G) \geq \frac{2}{r+1}n$. It was proved that for every graph $G$, $a(G) \geq \sum_{i=1}^{n}\frac{2}{d_i+1}$, where $d_1, \ldots, d_n$ is the degree sequence of $G$. Here, we conjecture that for every graph $G$ with $\delta(G) \geq 2$, LIF$(G) \geq \sum_{i=1}^{n}\frac{2}{d_i+1}$.