An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . , m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that e...

In this paper, we focus on the design of network topology to achieve fast information distribution. We present the information distribution performance of Borel Cayley graphs, a family of pseudo-random graphs, is far superior than that of other well-known graph families. To demonstrate the effectiveness of this pseudo-random approach, we compare the convergence speed of the average consensus pr...

Let G be a 2-connected graph with a toroidal rotation system given. An algorithm for constructing a straight line drawing with no crossings on a rectangular representation of the torus is presented. It is based on Read’s algorithm for constructing a planar layout of a 2-connected graph with a planar rotation system. It is proved that the method always works. The complexity of the algorithm is l...

As it is introduced by Bermond, Prennes, and Kodate and by Fragopoulou and Akl, some Cayley graphs, including most popular models for intercon-nection networks, admit a special automorphism, called complete rotation. Such an automorphism is often used to derive algorithms or properties of the underlying graph. For example, some optimal gossiping algorithms can be easily designed by using a comp...

An irreversible k-threshold process (also a k-neighbor bootstrap percolation) is a dynamic process on a graph where vertices change color from white to black if they have at least k black neighbors. An irreversible k-conversion set of a graph G is a subset S of vertices of G such that the irreversible k-threshold process starting with S black eventually changes all vertices of G to black. We sh...

We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree (∆) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on t...

A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2)and (5, 1)-colorable, and that a graph of genus γ is (χγ/(d + 1) + 4, d)-colorable, where χγ ...

We prove that every graph of girth at least five which admits an embedding in the Klein bottle is 3-colorable. This solves a problem raised by Woodburn, and complements a result of Thomassen who proved the same for projective planar and toroidal graphs. 15 July 1998, Revised 26 April 2004. To appear in J. Combin. Theory Ser. B. 1 Partially supported by NSF under Grant No. DMS-9623031 and by NSA...

A circulant of order n is a Cayley graph for the cyclic group Zn, and as such, admits a transitive action of Zn on its vertices. This paper concerns 2-cell embeddings of connected circulants on closed orientable surfaces. Embeddings on the sphere (the planar case) were classified by Heuberger (2003), and by a theorem of Thomassen (1991), there are only finitely many vertex-transitive graphs wit...

The gossip problem involves communicating a unique item from each node in a graph to every other node. We study the minimum time required to do this under the weakest model of parallel communication which allows each node to participate in just one communication at a time as either sender or receiver. We study a number of topologies including the complete graph, grids, hypercubes and rings. Dee...