Let G = (V, E) be a simple graph and H be a subgraph of G. Then G admits an H-covering, if every edge in E(G) belongs to at least one subgraph of G that is isomorphic to H. An (a, d)-H-antimagic total labeling of G is a bijection ƒ : V (G) ∪ E(G) → {1, 2, 3, ..., |V (G)| + |E(G)|} such that for all subgraphs H’ of G isomorphic to H, the H’ weights w(H’) =∑v∈V(H’) f(v) + =∑e∈E(H’) f(e) constitute an arithmetic progression {a, a + d, a + 2d, ..., a + (n − 1)d}, where a and d are positive integers and n is the number of subgraphs of G isomorphic to H. The labeling ƒ is called a super (a, d)-H-antimagic total labeling if ƒ (V (G)) = {1, 2, 3, ..., |V (G)|}. In [9], authors have posed an open problem to characterize the super (a, d)-G+ e-antimagic total labeling of the graph Gu[Sn], where n ≥ 3 and 4 ≤ d ≤ p+q + 2. In this paper, a partial solution to this problem is obtained.
Let G = (V,E) be a graph of order p and size q having no isolated vertices. A bijection f : V → {1, 2, 3, ..., p} is called a local edge antimagic labeling if for any two adjacent edges e = uv and e’ = vw of G, we have w(e) ≠ w(e’), where the edge weight w(e = uv) = f(u)+f(v) and w(e’) = f(v)+f(w). A graph G is local edge antimagic if G has a local edge antimagic labeling. The local edge antimagic chromatic number χ’lea(G) is defined to be the minimum number of colors taken over all colorings of G induced by local edge antimagic labelings of G. In this paper, we determine the local edge antimagic chromatic number for a friendship graph, wheel graph, fan graph, helm graph, flower graph, and closed helm.
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