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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Wulandari, Risma Yulina
Institut Teknologi Bandung
Electrical & Electronics Engineering

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D-antimagic labelings arising from completely separating systems Wulandari, Risma Yulina; Simanjuntak, Rinovia; Saputro, Suhadi Wido
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 13, No 2 (2025): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2025.13.2.2

Abstract

Let D be a non-empty subset of the distance set {0,1,…, diam(G)}. A graph G is D-antimagic if there exists a bijection f:V(G)→{1,2,…,|V(G)|} such that for every pair of distinct vertices x and y, wD(x) ≠ wD(y), where wD(x) = Σz∈ND(x)f(z) is the D-weight of x and ND(x) = {z|d(x,z)∈D} is the D-neighbourhood of x. It was conjectured that a graph G is D-antimagic if and only if each vertex in G has a distinct D-neighborhood. A completely separating system (CSS) in the finite set {1,2,…,n} is a collection ? of subsets of {1,2,…,n} in which for each pair a≠b∈{1,2,…,n}, there exist A,B∈? such that a∈A−B and b∈B−A.In this paper, we provide evidence to support the conjecture mentioned earlier by using Roberts' completely separating systems to define D-antimagic labelings for certain graphs. In particular, we show that if G and H are D-antimagic graphs with labelings constructed from Roberts' CSS, then the vertex-deleted subgraph, G−{v} and the vertex amalgamation of G and H are also D-antimagic. Additionally, we partially answer an open problem of Simanjuntak et al. (2021) by constructing {1}-antimagic labelings for some disjoint unions of paths.
Co-Authors Saputro, Suhadi Wido Simanjuntak, Rinovia