Let G = (V,E) be a graph of order n. Let f : V → {1, 2, . . . , n} be a bijection. For any vertex v ∈ V , the neighbor sum ∑u∈N(v) f(u) is called the weight of the vertex v and is denoted by w(v). If w(v) = k, (a constant) for all v ∈ V , then f is called a distance magic labeling with magic constant k. If the set of vertex weights forms an arithmetic progression {a, a+ d, a+2d, . . . , a+ (n− ...

Distance labeling is a preprocessing technique introduced by Peleg [Journal of Graph Theory, 33(3)] to speed up distance queries in large networks. Herein, each vertex receives a (short) label and, the distance between two vertices can be inferred from their two labels. One such preprocessing problem occurs in the hub labeling algorithm [Abraham et al., SODA’10]: the label of a vertex v is a se...

A vertex-distinguishing coloring of a graph G consists in an edge or a vertex coloring (not necessarily proper) of G leading to a labeling of the vertices of G, where all the vertices are distinguished by their labels. There are several possible rules for both the coloring and the labeling. For instance, in a set irregular edge coloring [5], the label of a vertex is the union of the colors of i...

For a simple graph G, a vertex labeling φ : V (G) → {1, 2, · · · , k} is called k-labeling. The weight of an edge xy in G, denoted by wπ(xy), is the sum of the labels of end vertices x and y, i.e. wφ(xy) = φ(x) + φ(y). A vertex k-labeling is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f , there is wφ(e) 6= wφ(f). The minimum k for which the g...

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 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)$...