Regular Graphs with Maximum Forest Number

Abstract

Punnim proved in [6] that if G is an r-regular graph of order n, then its forest number is at most c, where

$$c=\left\{ \begin{array} {ll} n-r+1 & {\rm if} \; r+1 \leq n \leq 2r-1,\\ \lfloor \frac{nr-2}{2(r-1)} \rfloor & {\rm if}\; n\geq 2r. \end{array} \right.$$

He also proved that the bound is sharp. Let ${\cal{R}}(r^{n};c)$ be the class of all r-regular graphs of order n. We prove in this paper that if $G, H\in{\cal{R}}(r^{n};c)$, then there exists a sequence of switchings σ ₁, σ ₂, …, σ _t such that for each i = 1, 2, …, t, $G^{\sigma_1\sigma_2\cdots\sigma_i}\in{\cal{R}}(r^n;c)$ and $H=G^{\sigma_1\sigma_2\cdots\sigma_t}$.