A vertex magic total labeling of a graph with v vertices and e edges is de ned as a one to one map taking the vertices and edges onto the integers v e with the property that the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex Properties of these labelings are studied It is shown how to construct labelings for several families o...

Let G=(V(G),E(G)) be a connected simple undirected graph with non empty vertex set V(G) and edge set E(G). For a positive integer k, by an edge irregular total k-labeling we mean a function f : V(G)UE(G) --> {1,2,...,k} such that for each two edges ab and cd, it follows that f(a)+f(ab)+f(b) is different from f(c)+f(cd)+f(d), i.e. every two edges have distinct weights. The minimum k for which G ...

An antimagic labeling of a graph G is bijection from the set edges E(G) to $$\{1,2,\ldots ,|E(G)|\}$$ , such that all vertex sums are pairwise distinct, where sum at u labels assigned incident u. A called when it has an labeling. Hartsfield and Ringel conjectured every simple connected other than $$K_2$$ conjecture remains open even for trees. Here, we prove caterpillars by means $$O(n \log n)$...

Let γ(G) be the domination number of a graphG. A graphG is dominationvertex-critical, or γ-vertex-critical, if γ(G − v) < γ(G) for every vertex v ∈ V(G). In this paper, we show that: Let G be a γ-vertex-critical graph and γ(G) = 3. (1) If G is of even order and K1,6-free, then G has a perfect matching; (2) If G is of odd order and K1,7-free, then G has a near perfect matching with only three ex...

Every integer is either even or odd, so we know that the polynomial f(x) = x(x− 1) 2 is integervalued on the integers, even though its coefficients are not in Z. Similarly, since every binomial coefficient ( k n ) is an integer, the polynomial ( x n ) = x(x− 1)...(x− n+ 1) n! must also be integervalued. These polynomials were used for polynomial interpolation as far back as the 17 century. Inte...

Abstract It is well-known that every Eulerian orientation of an 2 k -edge-connected undirected graph -arc-connected. A long-standing goal in the area has been to obtain analogous results for vertex-connectivity. Levit, Chandran and Cheriyan recently proved Levit et al. (2018) a hypercube dimension -vertex-connected. Here we provide elementary proof this result. We also show other families -regu...

Determining the size of minimum vertex cover of a graph G, denoted by β(G), is an NP-complete problem. Also, for only few families of graphs, β(G) is known. We study the size of minimum vertex cover in generalized Petersen graphs. For each n and k (n > 2k), a generalized Petersen graph P (n, k), is defined by vertex set {ui, vi} and edge set {uiui+1, uivi, vivi+k}; where i = 1, 2, . . . , n and...