On join product and local antimagic chromatic number of regular graphs

Abstract

Let G = (V, E) be a connected simple graph of order p and size q. A graph G is called local antimagic if G admits a local antimagic labeling. A bijection f : E → {1, 2, . . . , q} is called a local antimagic labeling of G if for any two adjacent vertices u and v, we have f, and E(u) is the set of edges incident to u. Thus, any local antimagic labeling induces a proper vertex coloring of G if vertex v is assigned the color f + (v). The local antimagic chromatic number, denoted χ la (G), is the minimum number of induced colors taken over local antimagic labeling of G. Let G and H be two vertex disjoint graphs. The join graph of G andIn this paper, we show the existence of non-complete regular graphs with arbitrarily large order, regularity and local antimagic chromatic numbers.

On local antimagic chromatic number of lexicographic product graphs

“…and that C m [O n ] is 2n-regular of order mn for m ≥ 3, n ≥ 1. Thus, together with those results in [5], we partially answer the following question. The following problems arise naturally.…”

“…In [5], the authors completely determined the local antimagic chromatic number of all connected regular graphs of order at most 8. Suppose m ≥ 3.…”

On local antimagic chromatic number of spider graphs

“…Let G = C 3 × K 2 and H be the octahedral graph. Figure 2 presents their local antimagic 3-colorings which are shown in [8]. It is easy to verify that G and H satisfy the conditions of Corollary 2.7, which implies that 3χ(H)…”