On local antimagic chromatic number of cycle-related join graphs (Q2214312)

In this paper, the authors discuss the local antimagic chromatic number of cycle-related join graphs. In Section \(1\), they give preliminary definitions and results regarding local antimagic edge labelings. The primary focus of this paper is based on finding the local antimagic chromatic number \(\chi_{la} (G)\) of graphs. In the second section, they investigate the bounds on graphs with an edge deleted or added for local antimagic graphs. In Section \(3\), the authors define the cycle related join graphs such as \(C_{m} \vee O_{n}\), \((C_{m} \vee O_{n})-e\), \(W_{m}\), \(W_{m}-e\), \(C_{m} \vee C_{n}\), \(C_{m} \vee K_{n}\), \(M_{2n}\), \(M_{6} \vee O_{2n}\) and \(G(m, n)\). Also they determine the local antimagic chromatic number for the above graphs. The authors successfully investigate \(\chi_{la} (W_{m}-e)\). They prove that \(W_{4}-e\) admits a local antimagic labeling \(f\) with \(c(f) = 3\) so that \(\chi_{la}(W_{4}-e) = 3\) if \(e \notin E(C_{4})\). They obtain a local antimagic labeling \(g\) for \(W_{4} - e\) with \(c(g) = 4\) if \(e \in E(C_{4})\). They prove that for even \(m \geq 6\), \(\chi_{la} (W_{m}-e) = 3\) where \(e \in E(W_{m})\). They also study the local antimagic number of \(W_{m}-e\) where \(e \in E(C_{m})\) and \(e\notin E(C_{m})\). \par The following are some salient features of this paper: This paper is a brilliant work done by the authors. It is written in precise language and with clarity of concepts. In this paper, several sufficient conditions for \(\chi_{la}(H) \leq \chi_{la}(G)\) are obtained, where \(H\) is obtained from \(G\) with a certain edge deleted or added. They obtain the exact value of local antimagic chromatic number of many cycle related join graphs. The use of tables is praiseworthy as they summarise different aspects of the proof. The given examples clearly validate the facts in this paper. The detailed references are appreciated. For those who do research in the area of labeling and coloring, this paper will be of great benefit when they read the results available in this paper. On the whole, the paper appears to be a creative piece of work.