The degree/diameter problem in maximal planar bipartite graphs

The (¿;D) (degree/diameter) problem consists of nding the largest possible number of vertices n among all the graphs with maximum degree ¿ and diameter D. We consider the (¿;D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (¿; 2) problem, the number of vertices is n = ¿+2; and for the (¿; 3) problem, n = 3¿¿1 if ¿ is odd and n = 3¿ ¿ 2 if ¿ is even. Then, we study the general case (¿;D) and obtain that an upper bound on n is approximately 3(2D + 1)(¿ ¿ 2)¿D=2¿ and another one is C(¿ ¿ 2)¿D=2¿ if ¿ D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (¿ ¿ 2)k if D = 2k, and 3(¿ ¿ 3)k if D = 2k + 1, for ¿ and D sufficiently large in both cases.Postprint (published version

The degree/diameter problem in maximal planar bipartite graphs

The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.Peer ReviewedPostprint (published version

4-labelings and grid embeddings of plane quadrangulations

AbstractA straight-line drawing of a planar graph G is a closed rectangle-of-influence drawing if for each edge uv, the closed axis-parallel rectangle with opposite corners u and v contains no other vertices. We show that each quadrangulation on n vertices has a closed rectangle-of-influence drawing on the (n−3)×(n−3) grid.The algorithm is based on angle labeling and simple face counting in regions. This answers the question of what would be a grid embedding of quadrangulations analogous to Schnyder’s classical algorithm for embedding triangulations and extends previous results on book embeddings for quadrangulations from Felsner, Huemer, Kappes, and Orden.A further compaction step yields a straight-line drawing of a quadrangulation on the (⌈n2⌉−1)×(⌈3n4⌉−1) grid. The advantage over other existing algorithms is that it is not necessary to add edges to the quadrangulation to make it 4-connected

Hamiltonicity and generalised total colourings of planar graphs

The total generalised colourings considered in this paper are colourings of graphs such that the vertices and edges of the graph which receive the same colour induce subgraphs from two prescribed hereditary graph properties while incident elements receive different colours. The associated total chromatic number is the least number of colours with which this is possible. We study such colourings for sets of planar graphs and determine, in particular, upper bounds for these chromatic numbers for proper colourings of the vertices while the monochromatic edge sets are allowed to be forests. We also prove that if an even planar triangulation has a Hamilton cycle H for which there is no cycle among the edges inside H, then such a graph needs at most four colours for a total colouring as described above. The paper is concluded with some conjectures and open problems.In part by the National Research Foundation of South Africa (Grant Numbers 90841, 91128).http://www.degruyter.com/view/j/dmgtam2016Mathematics and Applied Mathematic

Binary Labelings for Plane Quadrangulations and their Relatives

Graphs and Algorithm

Descomposicions de grafs en arbres.

Postprint (published version