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An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it 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 f+(x) = Σf(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, several sufficient conditions for χla(H) ≤ χla(G) are obtained, where H is obtained from G with a certain edge deleted or added. We then determined the exact value of the local antimagic chromatic number of many cycle-related join graphs.

An edge labeling of a graph G = (V , E) is said to be local antimagic if it 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 of x is f + (x) = ∑ e ∈ E (x) f (e) (E (x) is the set of 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. In this paper, several sufficient conditions to determine the local antimagic chromatic number of the join of graphs are obtained. We then determine the exact value of the local antimagic chromatic number of many join graphs. [ABSTRACT FROM AUTHOR]

In this note, we will focus on several applications on the Dirichlet's box principle in Discrete Mathematics lesson and number theory lesson. In addition, the main result is an innovative game on a triangular board developed by the authors. The game has been used in teaching and learning mathematics in Discrete Mathematics and some high schools in Hong Kong. (Contains 1 note.)

An edge labeling of a connected graph G = (V , E) is said to be local antimagic if it 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 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. In this paper, we show the existence of infinitely many bipartite and tripartite graphs with χ l a = 2 , 3. [ABSTRACT FROM AUTHOR]

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it 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 f+(x) = ∑f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. Let χ(G) be the chromatic number of G. In this paper, sharp upper and lower bounds of χla(G) for G with pendant vertices, and sufficient conditions for the bounds to equal, are obtained. Consequently, for k ≥ 1, there are infinitely many graphs with k ≥ χ(G) - 1 pendant vertices and χla(G) = k + 1. The work of this paper leads to conjecture and problem on graphs with pendant(s). [ABSTRACT FROM AUTHOR]