On local antimagic chromatic numbers of circulant graphs join with null graphs or cycles

Abstract

An edge labeling of a graph G = (V,E) is said to be local antimagic if there is a bijection f : E → {1,..., |E|} such that for any pair of adjacent vertices x and y, f ⁺(x) ≠ f ⁺(y), where the induced vertex label is f ⁺(x) = ? f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χ_la(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. For a bipartite circulant graph G, it is known that χ(G)=2 but χ_la(G) ≥ 3. Moreover, χ_la(C_n ∨ K₁)=3 (respectively 4) if n is even (respectively odd). Let G be a graph of order m ≥ 3. In [Affirmative solutions on local antimagic chromatic number, Graphs Combin., 36 (2020), 1337—1354], the authors proved that if m ≡ n (mod 2) with χ_la(G) = χ(G), m>n ≥ 4 and m ≥ n²/2, then χ_la(G∨O_n) = χ_la(G)+1. In this paper, we show that the conditions can be omitted in obtaining χ_la(G∨H) for some circulant graph G, and H is a null graph or a cycle. The local antimagic chromatic number of certain wheel related graphs are also obtained.