Extensions of the Scherk-Kemperman Theorem

Let Γ=(V,E) be a reflexive relation with a transitive automorphism group. Let F be a finite subset of V containing a fixed element v. We prove that the size of Γ(F) (the image of F) is at least

|F|+|Γ(v)|−|Γ⁻(v)∩F|.     

The enumeration of several classes of hexagonal systems

Ahexagonalsystemisafiniteconnectedplanegraphwithnocutvertexinwhicheveryinteriorfaceisboundedbyaregularhexagonofsidelengthone.AhexagonalsystemHissaidtobeacata-condensedhexagonalsystemifeaChvertexofHisontheboundaryofH;otherwise,apert-condensedhexagonalsystem.Chemistsusuallycallthembenzenoidsystems,andsomemathematicianscallthempolyhexgraphs.Chemistsareillterestedinthistakeofgraphsandtheenumerationofthemsincetheyrepreselltthecarbonatomskeletongraphsofbenzenoidhydrocarbons[2--31.Ontheotherhand,th…     

Spanning maximal planar subgraphs of random graphs

We study the threshold for the existence of a spanning maximal planar subgraph in the random graph G_{n, p} . We show that it is very near p = 1/n^? We also discuss the threshold for the existence of a spanning maximal outerplanar subgraph. This is very near p = 1/n^½.     

Counting labeled claw-free cubic graphs by connectivity

Claw-free cubic graphs are counted with given connectedness and order. Tables are provided for claw-free cubic graphs with given connectedness. This builds on methods for counting general cubic graphs by connectivity previously developed by Chae, Palmer, and Robinson, and on the earlier enumeration of all claw-free cubic graphs by McKay, Palmer, Read, and Robinson.     

Counting colored planar maps: Algebraicity results

We address the enumeration of properly q-colored planar maps, or more precisely, the enumeration of rooted planar maps M weighted by their chromatic polynomial χM(q) and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when q≠0,4 is of the form 2+2cos(jπ/m), for integers j and m. This includes the two integer values q=2 and q=3. We extend this to planar maps weighted by their Potts polynomial PM(q,ν), which counts all q-colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper q-colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two “catalytic” variables. To our knowledge, this is the first time such equations are being solved since Tutte?s remarkable solution of properly q-colored triangulations.     

A universal planar graph under the minor relation

We construct an infinite planar graph that contains every planar graph as a minor. © 1999 John Wiley & Sons, Inc. J. Graph Theory 32: 191–206, 1999     

Degree distribution in random planar graphs

We prove that for each k?0, the probability that a root vertex in a random planar graph has degree k tends to a computable constant dk, so that the expected number of vertices of degree k is asymptotically dkn, and moreover that k_∑dk=1. The proof uses the tools developed by Giménez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=k_∑dkwk. From this we can compute the dk to any degree of accuracy, and derive the asymptotic estimate dk∼c⋅k−1/2qk for large values of k, where q≈0.67 is a constant defined analytically.     

On types of growth for graph-different permutations

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity “within a given type.”     

Bipolar neutrosophic planar graphs

Fuzzy graph theory is used for solving real-world problems in different fields, including theoretical computer science, engineering, physics, combinatorics and medical sciences. In this paper, we present conepts of bipolar neutrosophic multigraphs, bipolar neutrosophic planar graphs, bipolar neutrosophic dual graphs, and study some of their related properties. We also describe applications of bipolar neutrosophic graphs in road network and electrical connections.     

Intersection reverse sequences and geometric applications

Pinchasi and Radoi?i? [On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-crossing C₄ has O(n^3/2logn) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n^3/2logn) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.