Results 1  10
of
36
Random triangulations of planar points sets
"... Given a set S of n points in the plane, a triangulation is a maximal crossingfree geometric graph on S (in a geometric graph the edges are realized by straight line segments). Here we consider random triangulations, where “random ” refers to uniformly at random from the set of all triangulations of ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
Given a set S of n points in the plane, a triangulation is a maximal crossingfree geometric graph on S (in a geometric graph the edges are realized by straight line segments). Here we consider random triangulations, where “random ” refers to uniformly at random from the set of all triangulations of S. We are primarily interested in the degree sequences of such random triangulations.
Counting Triangulations of Planar Point Sets
"... We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30 n. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of 43^n for the problem. More ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30 n. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of 43^n for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossingfree) straightline graphs on a given point set. Specifically, we derive new upper bounds for the number of planar graphs (O ∗ (239.4 n)), spanning cycles (O ∗ (70.21 n)), spanning trees (160 n), and cyclefree graphs (O ∗ (202.5 n)).
The Penrose Polynomial of a Plane Graph
 Math. Ann
, 1996
"... this paper to study the Penrose polynomial of an arbitrary connected plane graph and, in particular, to generalize the Penrose formulae to this case. In section 2, we give the definition of P (G; ) and present an expression via cycles and cocycles of the graph G. In section 3 we look at some interes ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
this paper to study the Penrose polynomial of an arbitrary connected plane graph and, in particular, to generalize the Penrose formulae to this case. In section 2, we give the definition of P (G; ) and present an expression via cycles and cocycles of the graph G. In section 3 we look at some interesting coefficients of P (G; ). Section 4 is devoted to the evaluation of P (G; ) at integral , and section 5 deals with yet another formula of Penrose involving the rotation signs around the vertices (see [2]). The graphs considered in this paper are finite and may have loops and multiple edges. We use the standard notation of graph theory. For all terms not defined see e.g. [1,4,14,22]. 2. Definition of P (G; )
The Grötzsch Theorem for the hypergraph of maximal cliques
 Electron. J. Combin
, 1999
"... In this paper, we extend the Grotzsch Theorem by proving that the clique hypergraph of every planar graph is 3colorable. We also extend this result to list colorings by proving that for every planar or projective planar graph G. Finally, 4choosability is established for the class of loc ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
In this paper, we extend the Grotzsch Theorem by proving that the clique hypergraph of every planar graph is 3colorable. We also extend this result to list colorings by proving that for every planar or projective planar graph G. Finally, 4choosability is established for the class of locally planar graphs on arbitrary surfaces.
Graph domination, coloring and cliques in telecommunications
 Handbook of Optimization in Telecommunications, pages 865–890. Spinger Science + Business
, 2006
"... This paper aims to provide a detailed survey of existing graph models and algorithms for important problems that arise in different areas of wireless telecommunication. In particular, applications of graph optimization problems such as minimum dominating set, minimum vertex coloring and maximum cliq ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
This paper aims to provide a detailed survey of existing graph models and algorithms for important problems that arise in different areas of wireless telecommunication. In particular, applications of graph optimization problems such as minimum dominating set, minimum vertex coloring and maximum clique in multihop wireless networks are discussed. Different forms of graph domination have been used extensively to model clustering in wireless ad hoc networks. Graph coloring problems and their variants have been used to model channel assignment and scheduling type problems in wireless networks. Cliques are used to derive bounds on chromatic number, and are used in models of traffic flow, resource allocation, interference, etc. In this paper we survey the solution methods proposed in the literature for these problems and some recent theoretical results that are relevant to this area of research in wireless networks.
Some Remarks on the Odd Hadwiger's Conjecture
, 2005
"... We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are twocolored in such a way that the edges within the trees are bichromatic, but the edges between tre ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are twocolored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)colorable. This is substantially stronger than the wellknown conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd Kkminor is ck √ log kcolorable. However, it is not known if there exists an absolute constant c such that any graph with no odd Kkminor is ckcolorable.
Tutte's 3Flow Conjecture and Matchings in Bipartite Graphs
, 2001
"... Tutte's 3flow conjecture is restated as the problem of nding an orientation of the edges of a 4edgeconnected, 5regular graph G, for which the outflow at each vertex is +3 or 3. The induced equipartition of the vertices of G is called mod 3orientable. We give necessary and sufficient condi ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Tutte's 3flow conjecture is restated as the problem of nding an orientation of the edges of a 4edgeconnected, 5regular graph G, for which the outflow at each vertex is +3 or 3. The induced equipartition of the vertices of G is called mod 3orientable. We give necessary and sufficient conditions for the existence of mod 3orientable equipartitions in general 5regular graphs, in terms of (i) a perfect matching of a bipartite graph derived from the equipartition and (ii) the size of cuts in G. Also, we give a polynomial time algorithm for testing whether an equipartition is mod 3orientable.