On local antimagic chromatic number of spider graphs

Abstract

Let G = (V, E) be a connected simple graph. A bijection f :holds for any two adjacent vertices u and v, where f + (u) = e∈E(u) f (e) and E(u) is the set of edges incident to u. A graph G is called local antimagic if G admits at least a local antimagic labeling. The local antimagic chromatic number, denoted χ la (G), is the minimum number of induced colors taken over local antimagic labelings of G. Let G and H be two disjoint graphs. The graph G[H] is obtained by the lexicographic product of G and H. In this paper, we obtain sufficient conditions for χ la (G[H]) ≤ χ la (G)χ la (H). Consequently, we give examples of G and H such that χ la (G[H]) = χ(G)χ(H), where χ(G) is the chromatic number of G. We conjecture that (i) there are infinitely many graphs G and H such that χ la (G[H]) = χ la (G)χ la (H) = χ(G)χ(H), and (ii) for k ≥ 1, χ la (G[H]) = χ(G)χ(H) if and only if χ(G)χ(H) = 2χ(H) + ⌈ χ(H) k ⌉, where 2k + 1 is the length of a shortest odd cycle in G.

On bridge graphs with local antimagic chromatic number 3

“…, a s ), is a tree formed by identifying an end-vertex, called the core vertex, of each path of length a i for each 1 ≤ i ≤ s, where 1 ≤ a 1 ≤ • • • ≤ a s . In [7], the authors obtained many sufficient conditions so that χ la (Sp(a 1 , a 2 , . .…”

“…So we only consider s ≥ 3. From the proof of [7,Theorem 2.11], the core vertex has the induced color q, all vertices of degree 2 are labelled by q or q + 1, where q is the size of the spider graph. Now we merge all pendant vertices.…”

“…Local antimagic labeling was firstly proposed by Arumugam et al [1] in 2017. Since then, many scholars have studied the local antimagic chromatic numbers of different types of graphs, such as Nazulaa [9], Lau [4,5,6,7,8], Shaebani [10] and so on. Recently, Lau et al characterized s-bridge graphs with local antimagic chromatic number 2 in [8].…”

On local antimagic chromatic number of a corona product graph

“…Consequently, we showed that χ la (f n • O m ) = m(2n + 1) + 2 for n ≥ 2, m = 1. Interested readers may refer to [9][10][11][12] for local antimagic chromatic number of graphs with pendant edges.…”

Affirmative Solutions on Local Antimagic Chromatic Number

“…Thus, Sp(2 [n] ) is the spider graph with n legs of length 2. A spider graph is also know as a one point union of at least three paths at an end vertex (see [4]). Note that Sp(2 [n] ) is known as the amalgamation of n copies of P 3 at a pendant vertex, denoted Amal(P 3 , x, n) in [5], where x is the degree n vertex of Sp(2 [n] ).…”