For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-textit{magic}  if there exists a labeling $f:E(G) rightarrow A-{0}$ such that, the vertex labeling $f^+$  defined as $f^+(v) = sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$  the group of integers modulo $k...

The generalized prism πG of G is the graph consisting of two copies of G, with edges between the copies determined by a permutation π acting on the vertices of G. We define a generalized Cartesian product G π H that corresponds to the Cartesian product G H when π is the identity, and the generalized prism when H is the graph K2. Burger, Mynhardt and Weakley [On the domination number of prisms o...

A generalized prism, or more specifically an (0, j)-prism of order 2n (where n is even) is a cubic graph consisting of two cycles u The question of factorization of complete bipartite graphs into (0, j)-prisms was completely settled by the author and S. Cichacz. Some partial results have also been obtained by them and P. Kovar and S. Dib. We will present a decomposition for another class of (0,...

A permutation graph G of a graph G (or generalized prism) is obtained by taking two disjoint copies of G and adding an arbitrary matching between the copies. For the parameters diameter, radius, average distance, connectivity and edge-connectivity, we compare the values of the parameter for G and G. In particular we show that if G has no isolates and is not 2Kk for k odd, then there exists a pe...

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets Π1, . . . ,Πk, where Πi, i ∈ [k], is an i-packing. The following conjecture is posed and studied: if G is a subcubic graph, then χρ(S(G)) ≤ 5, where S(G) is the subdivision of G. The conjecture is proved for all generalized prisms of cycles. To get this result it ...

‎a $(p‎,‎q)$ graph $g$ is said to have a $k$-odd mean‎ ‎labeling $(k ge 1)$ if there exists an injection $f‎ : ‎v‎ ‎to {0‎, ‎1‎, ‎2‎, ‎ldots‎, ‎2k‎ + ‎2q‎ - ‎3}$ such that the‎ ‎induced map $f^*$ defined on $e$ by $f^*(uv) =‎ ‎leftlceil frac{f(u)+f(v)}{2}rightrceil$ is a‎ ‎bijection from $e$ to ${2k - ‎‎‎1‎, ‎2k‎ + ‎1‎, ‎2k‎ + ‎3‎, ‎ldots‎, ‎2‎ ‎k‎ + ‎2q‎ - ‎3}$‎. ‎a graph that admits $k$...

The prism over a graph G is the Cartesian productG K2 of G with the complete graph K2. If the prism over G is hamiltonian, we say that G is prism-hamiltonian. We prove that triangulations of the plane, projective plane, torus, and Klein bottle are prism-hamiltonian. We additionally show that every 4-connected triangulation of a surface with sufficiently large representativity is prism-hamiltoni...

The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we give formula for Merrifield-Simmons and Hosoya indices of some classes of cartesian product of two graphs K{_2}×H, where H is a path graph P{_n}, cyclic graph C{_n}, or star gra...

The prism graph is the dual of the complete graph on five vertices with an edge deleted, K5\e. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac’s infinite families of graphs and four inf...

Paul Erdos defined the concept of coprime graph and studied about cycles in coprime graphs. In this paper this concept is generalized and a new graph called Generalized coprime graph is introduced. Having observed certain basic properties of the new graph it is proved that the chromatic number and the clique number of some generalized coprime graphs are equal.