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2015 | 138 | 1 | 23-42
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Equations relating factors in decompositions into factors of some family of plane triangulations, and applications (with an appendix by Andrzej Schinzel)

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Let 𝓟 be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in dual terms) that every graph P ∈ 𝓟 has a decomposition into factors P₀, P₁, P₂ (indexed by elements of the cyclic group Q = {0,1,2}) such that every factor $P_{q}$ consists of two induced paths of the same length M(q), and K(q) - 1 induced cycles of the same length 2M(q). For q ∈ Q, we define an integer S⁺(q) such that the vector (K(q),M(q),S⁺(q)) determines the graph P (if P is simple) uniquely up to orientation-preserving isomorphism. We establish arithmetic equations that will allow calculating (K(q+1),M(q+1),S⁺(q+1)) from (K(q),M(q),S⁺(q)), q ∈ Q. We present some applications of these equations. The set {(K(q),M(q),S⁺(q)): q ∈ Q} is called the orbit of P. If P has a one-point orbit, then there is an orientation-preserving automorphism σ such that $σ(P_{i}) = P_{i+1}$ for every i ∈ Q (where P₃ = P₀). We characterize one-point orbits of graphs in 𝓟. It is known that every graph in 𝓟 has an even order. We prove that if P is of order 4n + 2, n ∈ ℕ, then it has two disjoint induced trees of the same order, which are equitable 2-colorable and together cover all vertices of P.
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  • Institute of Mathematics and Cybernetics, University of Economics, Komandorska 118/120, 53-345 Wrocław, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-1-2
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