The discrete and continuous graph labeling problem are discussed. A basis for the continuous graph labeling problem is presented, in which an explicit connection between the discrete and continuous problems is made. The need for this basis is argued by noting conditions which must be satisfied before solutions can be pursued in a formal manner. Several cooperative solution algorithms based on t...

For a graph G and any two vertices u and v in G, let d(u, v) denote the distance between u and v and let diam(G) be the diameter of G. A multilevel distance labeling (or radio labeling) for G is a function f that assigns to each vertex of G a positive integer such that for any two distinct vertices u and v, d(u, v)+ | f(u) − f(v) |≥ diam(G) + 1. The largest integer in the range of f is called t...

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An antimagic labeling of a directed graph D with n vertices and m arcs is a bijection from the set of arcs of D to the integers {1, . . . ,m} such that all n oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. An undirected graph G is said to have an antimagic orientation i...

An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . ,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [4] conjectured tha...

An antimagic labeling of an undirected graph G with n vertices and m edges is a bijection from the set of edges of G to the integers {1, . . . ,m} such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labeling. In [6], Hartsfield and Ringel conjectured that every si...

Given a directed graph G = (V, A) (with n = |V | and m = |A|) with a length function : A → R + and a pair of vertices s, t, a distance oracle returns the distance dist(s, t) from s to t. A labeling algorithm [18] implements distance oracles in two stages. The preprocessing stage computes a label for each vertex of the input graph. Then, given s and t, the query stage computes dist(s, t) using o...

An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . ,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. A conjecture of Ringel (see [4]) states that every connected graph, but K2,...

A graph G is called a sum graph if there is a so-called sum labeling of G, i.e. an injective function l : V (G) → N such that for every u, v ∈ V (G) it holds that uv ∈ E(G) if and only if there exists a vertex w ∈ V (G) such that l(u) + l(v) = l(w). We say that sum labeling l is minimal if there is a vertex u ∈ V (G) such that l(u) = 1. In this paper, we show that if we relax the conditions (ei...

An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . , m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [5] conjectured th...