The decycling number of cubic planar graphs

• Authors: Narong Punnim
• Venue: CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
• Year: 2005

Bau and Beineke [2] asked the following questions: 1 Which cubic graphs G of order 2n have decycling number φ(G) = = ⌈nċċ/ċ⌉? 2 Which cubic planar graphs G of order 2n have decycling number φ(G) = ⌈nċċ/ċ⌉? We answered the first question in [10]. In this paper we prove that if P(3ċn) is the class of all connected cubic planar graphs of order 2n and φ (P(3ċn)) = {φ(G) : G ∈ P(3ċn)}, then there exist integers an and bn such that there exists a graph G ∈ P(3ċn) with φ(G) = c if and only if c is an integer satisfying an ≤ c ≤ bn. We also find all corresponding integers an and bn. In addition, we prove that if P (3ċn; φ = ⌈nċċ/ċ⌉) is the class of all connected cubic planar graphs of order 2n with decycling number ⌈nċċ/ċ⌉ and |G., G. ∈ P(3ċn; φ = ⌈nċċ/ċ⌉, then there exists a sequence of switchings σ., σ., ..., σt such that for every i = 1, 2, ..., t- 1, Gσ1σ2....σi ∈ P (3ċn; φ = ⌈nċċ/ċ⌉) and G. = G.σ1σ2....σt