We obtain some improved upper and lower bounds on the oriented chromatic number for different classes of products of graphs.

Fault tolerance and transmission delay of networks are important concepts in network design. The notions are strongly related to connectivity and diameter of a graph, and have been studied by many authors. Wide diameter of a graph combines studying connectivity with the diameter of a graph. Diameter with width k of a graph G is defined as the minimum integer d for which there exist at least k i...

A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and denoted by χ=(G). It is interesting to note that if a graph G is equitably k-colorable, it does not imply that it is equitab...

Graham and Pollak showed that the vertices of any connected graph G can be assigned t-tuples with entries in {0, a, b}, called addresses, such that the distance in G between any two vertices equals the number of positions in their addresses where one of the addresses equals a and the other equals b. In this paper, we are interested in determining the minimum value of such t for various families...

Let H be a bipartite graph and let Gn be the Mycielski graph with χ(G) = n, n ≥ 4. Then the chromatic number of the strong product of Gn by H is at most 2n− 2. We use this result to show that there exist strong products of graphs in which a projection of a retract onto a factor is not a retract of the factor. We also show that in the Cartesian product of graphs G and H, any triangles of G trans...

Our first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product. Our second main result gives a presentation for the inverse hull of such a graph product. We then specialise to the ca...

A finite graph product is the fundamental group of a finite graph of groups. Finite graph products with finite cyclic edge groups are shown to inherit a solvable conjugacy problem from their vertex groups under certain conditions, of which the most important is that all the edge group generators in each vertex group are powers of a common central element.

The Hadwiger number (G) of a graph G is the largest integer h such that the complete graph on h nodes Kh is a minor of G. Equivalently, (G) is the largest integer such that any graph on at most (G) nodes is a minor ofG. The Hadwiger’s conjecture states that for any graph G, (G) (G), where (G) is the chromatic number of G. It is well-known that for any connected undirected graph G, there exists ...

A graph G is an efficient open domination graph if there exists a subset D of V (G) for which the open neighborhoods centered in vertices of D form a partition of V (G). We completely describe efficient domination graphs among direct, lexicographic and strong products of graphs. For the Cartesian product we give a characterization when one factor is K2 and some partial results for grids, cylind...

A stable set in a graph G is a set of pairwise non-adjacent vertices. The independence polynomial of G is I(G;x) = s0+s1x+s2x 2 +...+sαx α , where α = α(G) is the cardinality of a maximum stable of G, while sk equals the number of stable sets of size k in G (Gutman and Harary, 1983). Hamidoune, 1990, showed that for every claw-free graph G (i.e., a graph having no induced subgraph isomorphic to...