Decompositions and packings in truncated triangulations - Michael Muzheve (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2026)
We study decompositions and packings in truncated triangulations GT△ obtained from simple connected plane graphs G with minimum degree two. We show GT△ is a 3-connected cubic planar graph with at least 2|E(G)|² - 2|E(G)| + 1 perfect matchings, a Λ-factor, and can be decomposed into a union of C₆'s and K₂'s if G is bipartite. Additionally, we show that GT△ is hamiltonian if G is bipartite with a dominating path P satisfying, for any e = xy ∉ E(P) exactly one of x and y is in V(P). We also prove a result giving necessary and sufficient conditions for the hamiltonicity of GT△. Additional results include showing that a truncated triangulation of a cubic plane bipartite graph G has a hamiltonian cycle that separates specific faces of GT△ if and only if the triangulation G△ has an A-trail.