We prove that, for a certain positive constant a and for an infinite set of values of n, the number of nonisomorphic face 2-colourable triangular embeddings of the complete graph Kn in an orientable surface is at least nan 2 . AMS classification: 05B07, 05C10.

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed v...

We show that the problem of counting number flats size k for a cycle matroid complete graph is equivalent to partitions an integer into triangular numbers. In addition, we give some values such there no flat in order k. Finally, minimum bound values, k, which are given matroid.

A graph G is (a, b)-choosable if for any color list of size a associated with each vertex, one can choose a subset of b colors such that adjacent vertices are colored with disjoint color sets. This paper proves that for any integer m ≥ 1, every finite triangle-free induced subgraph of the triangular lattice is (5m, 2m)-choosable.

Geographic routing is a routing paradigm, which uses geographic coordinates of network nodes to determine routes. Greedy routing, the simplest form of geographic routing forwards a packet to the closest neighbor towards the destination. A greedy embedding is a embedding of a graph on a geometric space such that greedy routing always guarantees delivery. A Schnyder drawing is a classical way to ...

A dominating set of a graph G is vertices that contains at least one endpoint every edge on the graph. The domination number order minimum G. (t, r) broadcast generalization in which broadcasting emits signals strength t decrease by 1 as they traverse each edge, and we require vertex receives cumulative signal r from its neighbors. In this paper, extend study to directed graphs. Our main result...

A dessin d'enfant, or dessin, is a bicolored graph embedded into Riemann surface, and the monodromy group an algebraic invariant of generated by rotations edges about black white vertices. rational billiards surface two dimensional that allows one to view path ball as continuous path. In this paper, we classify groups dessins associated triangular surfaces.

A k-chromatic graph G is called uniquely k-colorable if every k-coloring of the vertex set V of G induces the same partition of V into k color classes. There is an innnite class C of uniquely 4-colorable planar graphs obtained from the K 4 by repeatedly inserting new vertices of degree 3 in triangular faces. In this paper we are concerned with the well-known conjecture (see 6]) that every uniqu...

We investigate which entanglement resources allow universal measurement-based quantum computation via single-qubit operations. We find that any entanglement feature exhibited by the 2D cluster state must also be present in any other universal resource. We obtain a powerful criterion to assess the universality of graph states by introducing an entanglement measure which necessarily grows unbound...

We present an efficient algorithm that computes a manifold triangular mesh from a set of unorganized sample points in 3 . The algorithm builds on the observation made by several researchers that the Gabriel graph of the sample points provides a good surface description. However, this surface description is only one-dimensional. We associate the edges of the Gabriel graph with index 1 critical p...