An $L(h_1, h_2, \ldots, h_l)$-labelling of a graph $G$ is mapping $\phi: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that for $1\le i\le l$ and each pair vertices $u, v$ at distance $i$, we have $|\phi(u) - \phi(v)| \geq h_i$. The span $\phi$ the difference between largest smallest labels assigned to by $\phi$, $\lambda_{h_1, h_l}(G)$ defined as minimum over all h_l)$-labellings $G$. In this pap...

An Italian dominating function on a (di)graph G with vertex set V ( ) is f : → { 0 , 1 2 } such that every v ∈ = has an (in)neighbour assigned or two (in)neighbours 1. We complete the investigation of domination numbers Cartesian products directed cycles.

The dual notions of domination and packing in finite simple graphs were first extensively explored by Meir and Moon in [15]. Most of the lower bounds for the domination number of a nontrivial Cartesian product involve the 2-packing, or closed neighborhood packing, number of the factors. In addition, the domination number of any graph is at least as large as its 2-packing number, and the invaria...

The exact values of crossing numbers of the Cartesian products of four special graphs of order five with cycles are given and, in addition, all known crossing numbers of Cartesian products of cycles with connected graphs on five vertices are summarized.

In this article we introduced the isomorphism mapping between cartesian products of family of linear spaces [4]. Those products had been formalized by two different ways, i.e., the way using the functor [:X,Y:] and ones using the functor " product ". By the same way, the isomorphism mapping was defined between Cartesian products of family of linear normed spaces also. The notation and terminolo...

We study the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we prove that a product of graphs is the graph of a simple polytope if and only if its factors are. On the other hand, we provide a general construction of polytopal products of a polytopal graph by a non-polytopal graph.

An identifying code in a graph G is a set D of vertices such that the closed neighborhood of each vertex of the graph has a nonempty, distinct intersection with D. The minimum cardinality of an identifying code is denoted γ(G). Building upon recent results of Gravier, Moncel, and Semri, we show for n ≤ m that γID(Kn Km) = max{2m−n,m+⌊n/2⌋}. Furthermore, we improve upon the bounds for γ(G Km) an...