# Regular Graphs with Maximum Forest Number

• Avapa Chantasartrassmee
• Narong Punnim
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7033)

## 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 $$H=G^{\sigma_1\sigma_2\cdots\sigma_t}$$.

## Keywords

Planar Graph Nonnegative Integer Regular Graph Degree Sequence Graphical Sequence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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