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 ...

Let G be a toroidal graph without cycles of a fixed length k, and l (G) the list chromatic number of G. We establish tight upper bounds Contract grant sponsor: RGC; Contract grant number: CUHK4165/01E (to L. C.); Contract grant sponsor: NSFC; Contract grant number: 10771197 (to W. W.); Contract grant sponsor: ZJNSFC; Contract grant number: Z6090150 (to W. W.); Contract grant sponsor: NSC97-2115...

Endo [5] proved that every toroidal graph has a book embedding with at most seven pages. In this paper, we prove that every toroidal bipartite graph has a book embedding with at most five pages. In order to do so, we prove that every bipartite torus quadrangulation Q with n vertices admits two disjoint essential simple closed curves cutting the torus into two annuli so that each of the two annu...

A Schnyder wood is an orientation and coloring of the edges of a planar map satisfying a simple local property. We propose a generalization of Schnyder woods to graphs embedded on the torus with application to graph drawing. We prove several properties on this new object. Among all we prove that a graph embedded on the torus admits such a Schnyder wood if and only if it is an essentially 3conne...

Let c k = cr k (G) denote the minimum number of edge crossings when a graph G is drawn on an orientable surface of genus k. The (orientable) crossing sequence c 0 ; c 1 ; c 2 ; : : : encodes the trade-oo between adding handles and decreasing crossings. We focus on sequences of the type c 0 > c 1 > c 2 = 0; equivalently, we study the planar and toroidal crossing number of doubly-toroidal graphs....

The interplay between structural chemistry, graph theory and knot theory is a rich area for chemists and mathematicians alike. The founder of topology, Johann Listing, noted the possibility of chiral knots in 1847; a year later Pasteur published his landmark paper on the optical activity of ammonium tartrate crystals. Indeed, knot theory began following early studies on molecular isomerism. Mor...

We present the first survey of its kind on results at the intersection of topological graph theory and the game of Cops and Robbers, focusing on results, conjectures, and open problems for the cop number of a graph embedded on a surface. After a discussion on results for planar graphs, we consider graphs of higher genus. In 2001, Schroeder conjectured that if a graph has genus g, then its cop n...

In this note we study embeddings of Cayley graphs of right groups on surfaces. We characterize those right groups which have a toroidal but no planar Cayley graph, such that the generating system of the right group has a minimal generating system of the group as a factor.

In this paper we consider the spanning tree congestion for several families of graphs. We find the exact spanning tree congestion for toroidal meshes, Cm×Cn, and cylindrical meshes, Pm×Cn,. Also we find bounds for the spanning tree congestion of Qn, and a construction that gives the upper bound.