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 the same color. In this paper we investigate the existence of such colorings in surfaces. It is shown that a toroidal graph is (3; 2)and (5; 1)colorable, and that a graph of genus is ( =(d + 1) + 4; d)-colorable, where is the maximum chromatic number of a graph embe...

In this paper we study routing algorithms for one-to-one communication in multi-processors, whose interconnection networks have toroidal structure. A toroidal network is an n-dimensional rectangular mesh with additional edge-to-edge connections. Mathematically speaking, these networks are undirected graphs obtained by cartesian product of cycles. We show, that a broad class of routing algorithm...

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the integers {1, ..., q} such that all p vertex sums are pairwise distinct, where the vertex sum on a vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it has an anti-magic labeling. Hartsfield and Ringel [3] conjectured t...

Many manifold learning algorithms utilize graphs of local neighborhoods to estimate manifold topology. When neighborhood connections short-circuit between geodesically distant regions of the manifold, poor results are obtained due to the compromises that the manifold learner must make to satisfy the erroneous criteria. Also, existing manifold learning algorithms have difficulty unfolding manifo...

The problem on the Hamiltonicity of graphs is well studied in discrete algorithm and graph theory, because of its relation to traveling salesman problem (TSP). Starting with Tutte’s result, stating that every 4-connected planar graph is Hamiltonian, several researchers have studied the Hamiltonicity of graphs on surfaces. Extending Tutte’s technique, Thomassen proved that every 4-connected plan...

The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex x colored with color i, 1 ≤ i ≤ k, is adjacent to (i 1) vertices colored with each color j, 1 ≤ j ≤ i 1. In this paper we give bounds for the Grundy number of some graphs and Cartesian products of graphs. In particular, we determine an exact value of...