A note on bounded automorphisms of infinite graphs

LetX be a connected locally finite graph with vertex-transitive automorphism group. IfX has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic normal subgroup ofAUT(X) and the action ofAUT(X) onV(X) is imprimitive ifX is not finite. IfX has infinitely many ends, the group of bounded automorphisms itself is locally finite and periodic.     

On the weak reconstruction of Cartesian-product graphs

In this paper we reconstruct nontrivial connected Cartesian-product graphs from single vertex deleted subgraphs. We show that all one-vertex extensions of a given connected graph H, finite or infinite, to a nontrivial Cartesian product are isomorphic.     

Interference and traffic aware channel assignment in WiFi-based wireless mesh networks

Wireless mesh networks (WMN) typically employ mesh routers that are equipped with multiple radio interfaces to improve network capacity. The key aspect is to cleverly assign different channels (i.e., frequency bands) to each radio interface to form a WMN with minimum interference. The channel assignment must obey the constraints that the number of different channels assigned to a mesh router is at most the number of interfaces on the router, and the resultant mesh network is connected. This problem is known to be NP-hard. In this paper we propose a hybrid, interference and traffic aware channel assignment (ITACA) scheme that achieves good multi-hop path performance between every node and the designated gateway nodes in a multi-radio WMN network. ITACA addresses the scalability issue by routing traffic over low-interference, high-capacity links and by assigning operating channels in such a way to reduce both intra-flow and inter-flow interference. The proposed solution has been evaluated by means of both simulations and by implementing it over a real-world WMN testbed. Results demonstrate the validity of the proposed approach with performance increase as high as 111%.     

Cartesian products of directed graphs with loops

We show that every nontrivial finite or infinite connected directed graph with loops and at least one vertex without a loop is uniquely representable as a Cartesian or weak Cartesian product of prime graphs. For finite graphs the factorization can be computed in linear time and space.     

Antibandwidth of three-dimensional meshes

The antibandwidth problem is to label vertices of a graph G=(V,E) bijectively by 0,1,2,…,|V|−1 so that the minimal difference of labels of adjacent vertices is maximised. In this paper we prove an almost exact result for the antibandwidth of three-dimensional meshes. Provided results are extensions of the two-dimensional case and an analogue of the result for the bandwidth of three-dimensional meshes obtained by FitzGerald.     

On products of graphs and regular groups

A graphX is called a graphical regular representation (GRR) of a groupG if the automorphism group ofX is regular and isomorphic toG. Watkins and Nowitz have shown that the direct productG×H of two finite groupsG andH has aGRR if both factors have aGRR and if at least one factor is different from the cyclic group of order two. We give a new proof of this result, thereby removing the restriction to finite groups. We further show that the complementX′ of a finite or infinite graphX is prime with respect to cartesian multiplication ifX is composite and not one of six exceptional graphs.     

Tiled partial cubes

In the quest to better understand the connection between median graphs, triangle‐free graphs and partial cubes, a hierarchy of subclasses of partial cubes has been introduced. In this article, we study the role of tiled partial cubes in this scheme. For instance, we prove that almost‐median graphs are tiled and that tiled partial cubes are semi‐median. We also describe median graphs as tiled partial cubes without convex Q and extend an inequality for median graphs to a larger subclass of partial cubes. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 91–103, 2002