The Decycling Number of Cubic Planar Graphs

Abstract

Bau and Beineke [2] asked the following questions:

• 1 Which cubic graphs G of order 2n have decycling number $\phi(G)= \lceil\frac{n+1}{2}\rceil$?

• 1 Which cubic planar graphs G of order 2n have decycling number $\phi(G)=\lceil\frac{n+1}{2}\rceil$?

We answered the first question in [10]. In this paper we prove that if ${{\cal{P}}}(3^{2n})$ is the class of all connected cubic planar graphs of order 2n and ${\phi\left({\cal{P}}(3^{2n})\right)} = \left\{{\phi(G)\colon G\in {\cal{P}}}(3^{2n})\right\}$, then there exist integers a _n and b _n such that there exists a graph $G \in {\cal{P}}(3^{2n})$ with φ(G) = c if and only if c is an integer satisfying a _n  ≤ c ≤ b _n . We also find all corresponding integers a _n and b _n . In addition, we prove that if ${{\cal{P}}\left(3^{2n};\, \phi =\lceil\frac{n+1}{2}\rceil\right)}$ is the class of all connected cubic planar graphs of order 2n with decycling number $\lceil\frac{n+1}{2}\rceil$ and $G_1,\, G_2 \in {\cal{P}}\left(3^{2n};\, \phi =\lceil\frac{n+1}{2}\rceil\right)$, then there exists a sequence of switchings σ ₁, σ ₂, ..., σ _t such that for every i = 1, 2, ..., t − 1, $G_1^{\sigma_1\sigma_2\cdots\sigma_i} \in {\cal{P}}\left(3^{2n};\, \phi =\lceil\frac{n+1}{2}\rceil\right)$ and $G_2 = G_1^{\sigma_1\sigma_2\cdots\sigma_t}$.

This work was carried out with financial support by The Thailand Research Fund.