Abstract
An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.
Highlights
The notion of antimagic labeling of a graph G was introduced in Hartsfield and Ringel’s book Pearls in Graph Theory [1] as a bijection f :E( G ) → {1, . . . , e} such that the weights (w( x ) = ∑ xy∈E(G) f) distinguish all vertices
Hartsfield and Ringel [1] conjectured that every connected graph other than K2 admits antimagic labeling in this seminal work
In 2017, Arumugam et al [11] and Bensmail et al [12] independently introduced a weaker notion of antimagic labeling, called the local antimagic labeling, where only adjacent vertices must be distinguished
Summary
In 2017, Arumugam et al [11] and Bensmail et al [12] independently introduced a weaker notion of antimagic labeling, called the local antimagic labeling, where only adjacent vertices must be distinguished Both sets of authors conjectured that any connected graph other than K2 admits a local antimagic labeling. It is clear that if a graph contains two vertices having the same D-neighborhood set, the graph does not admit a D-antimagic labeling. It is clear that ∼ D is an equivalence relation, and Conjecture 1 can be rewritten as: “A graph admits a D-antimagic labeling if and only if its vertex set partition defined by ∼ D contains only singletons”. Examples of disjoint union of non-( D, r )regular graphs that are neither vertex-transitive nor edge-transitive but admit D-antimagic labelings are presented . We realize that Conjecture 1, if true, will be laborious to prove, and we propose several open problems that hopefully are more feasible to solve
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