The Linear Arboricity of Planar Graphs with Maximum Degree at Least Five

Let G be a planar graph with maximum degree ∆ ≥ 5. It is proved that la(G) = ∆(G)/2 if (1) any 4-cycle is not adjacent to an i-cycle for any i ∈ {3, 4, 5} or (2) G has no intersecting 4-cycles and intersecting i-cycles for some i ∈ {3, 6}

On the Geometric Thickness of 2-Degenerate Graphs

A graph is 2-degenerate if every subgraph contains a vertex of degree at most 2. We show that every 2-degenerate graph can be drawn with straight lines such that the drawing decomposes into 4 plane forests. Therefore, the geometric arboricity, and hence the geometric thickness, of 2-degenerate graphs is at most 4. On the other hand, we show that there are 2-degenerate graphs that do not admit any straight-line drawing with a decomposition of the edge set into 2 plane graphs. That is, there are 2-degenerate graphs with geometric thickness, and hence geometric arboricity, at least 3. This answers two questions posed by Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004]

A Structural Property of Trees with an Application to Vertex-Arboricity

We provide a structural property of trees, which is applied to show that if a plane graph G contains two edge-disjoint spanning trees, then its dual graph G⁎ has the vertex-arboricity at most 2. We also show that every maximal plane graph of order at least 4 contains two edge-disjoint spanning trees

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure