An antimagic labeling of a directed graph D with n vertices and m arcs is bijection from the set to integers { 1 , … } such that all oriented vertex sums are pairwise distinct, where an sum labels entering minus leaving it. A has orientation if it admits labeling. Hefetz, Mütze, Schwartz conjectured every connected orientation. In this paper, we show bipartite no degree 0 or 2 minimum at least 33

An antimagic labeling of a graph G is bijection f:E(G)?{1,…,|E(G)|} such that the weights w(x)=?y?xf(y) distinguish all vertices. A well-known conjecture Hartsfield and Ringel (1990) every connected other than K2 admits an labeling. For set distances D, D-antimagic f:V(G)?{1,…,|V(G)|} weight?(x)=?y?ND(x)f(y) distinct for each vertex x, where ND(x)={y?V(G)|d(x,y)?D} D-neigbourhood x. If ND(x)=r,...

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, ..., q} such that the vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an antimagic labeling. Hartsfield and Ringel conject...

Let be a connected graph with vertex set and edge . A bijection from to the is labeling of The called rainbow antimagic if for any two in path , where Rainbow coloring which has labeling. Thus, every induces G weight color connection number smallest colors all colorings denoted by In this study, we studied have an exact value joint product

In this paper we introduce a new type of graph labeling, the (a, d)vertex-antimagic total labeling, which is a generalization of several other types of labelings. A connected graph G(V, E) is said to be (a, d)-vertex-antimagic total if there exist positive integers a, d and a bijection λ : V ∪ E → {1, 2, . . . , |V | + |E|} such that the induced mapping gλ : V → W is also a bijection, where W =...

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)$...

A connected graph G = (V, E) is said to be (a, d)antimagic, for some positive integers a and d, if its edges admit a labeling by all the integers in the set {1, 2, . . . , |E(G)|} such that the induced vertex labels, obtained by adding all the labels of the edges adjacent to each vertex, consist of an arithmetic progression with the first term a and the common difference d. Mirka Miller and Mar...

Given that a graph G = (V, E). By an edge-antimagic vertex labeling of graph, we mean assigning labels on each under the label function f : V → {1, 2, . , |V (G)|} such associated weight edge uv ∈ E(G), namely w(xy) f(x) + f(y), has distinct weight. A path P in vertex-labeled is said to be rainbow if for every two edges xy, x′y ′ E(P) satisfies ̸= w(x ′y ). The called antimagic vertices x and y ...

The local antimagic total vertex labeling of graph G is a that every vertices and edges label by natural number from 1 to such two adjacent has different weights, where sum the labels all incident vertex. If start smallest then edge so kind coloring called super labeling. That induces for v, weight w(v) color v. minimum colors obtained chromatic G, denoted χlsat(G). In this paper, we consider G...