A total labeling of a graph G = (V, E) is said to be local total antimagic if it is a bijection f : V ∪ E → {1, . . . , |V | + |E|} such that adjacent vertices, adjacent edges, and incident vertex and edge have distinct induced weights where the induced weight of a vertex v, w f (v) =f (e) with e ranging over all the edges incident to v, and the induced weight of an edge uv is w f (uv) = f (u) + f (v). The local total antimagic chromatic number of G, denoted by χ lt (G), is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of G. In this paper, we first obtained general lower and upper bounds for χ lt (G) and sufficient conditions to construct a graph H with k pendant edges and χ lt (H) ∈ {∆(H) + 1, k + 1}. We then completely characterized graphs G with χ lt (G) = 3. Many families of (disconnected) graphs H with k pendant edges and χ lt (H) ∈ {∆(H) + 1, k + 1} are also obtained.

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, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pendants is obtained. Conjecture 3.1 in [Affirmative solutions on local antimagic chromatic number (2018), submitted] is completely solved. The exact value of the local antimagic chromatic number of many families of graphs with cutvertices are also determined. Consequently, we partially answered Problem 3.1 in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33 (2017) 275-285.].

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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) =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 give counterexamples to the lower bound of χ la (G ∨ O 2 ) that was obtained in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33 : 275 -285 (2017)]. A sharp lower bound of χ la (G ∨ O n ) and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs.

A (p, q, r)-board that has pq + pr + qr squares consists of a (p, q)-, a (p, r)-, and a (q, r)-rectangle. Let S be the set of the squares. Consider a bijection f : S → [1, pq + pr + qr]. Firstly, for 1 ≤ i ≤ p, let x i be the sum of all the q + r integers in the i-th row of the (p, q + r)-rectangle. Secondly, for 1 ≤ j ≤ q, let y j be the sum of all the p + r integers in the j-th row of the (q, p + r)-rectangle. Finally, for 1 ≤ k ≤ r, let z k be the the sum of all the p + q integers in the k-th row of the (r, p + q)-rectangle. Such an assignment is called a (p, q, r)-design if {x i : 1 ≤ i ≤ p} = {c 1 } for some constant c 1 , {y j : 1 ≤ j ≤ q} = {c 2 } for some constant c 2 , and {z k : 1 ≤ k ≤ r} = {c 3 } for some constant c 3. A (p, q, r)-board that admits a (p, q, r)-design is called (1) Cartesian tri-magic if c 1 , c 2 and c 3 are all distinct; (2) Cartesian bi-magic if c 1 , c 2 and c 3 assume exactly 2 distinct values; (3) Cartesian magic if c 1 = c 2 = c 3 (which is equivalent to supermagic labeling of K(p, q, r)). Thus, Cartesian magicness is a generalization of magic rectangles into 3-dimensional space. In this paper, we study the Cartesian magicness of various (p, q, r)-board by matrix approach involving magic squares or rectangles. In Section 2, we obtained various sufficient conditions for (p, q, r)-boards to admit a Cartesian tri-magic design. In Sections 3 and 4, we obtained many necessary and (or) sufficient conditions for various (p, q, r)-boards to admit (or not admit) a Cartesian bi-magic and magic design. In particular, it is known that K(p, p, p) is supermagic and thus every (p, p, p)-board is Cartesian magic. We gave a short and simpler proof that every (p, p, p)-board is Cartesian magic. Keywords 3-dimensional boards, Cartesian tri-magic, Cartesian bi-magic, Cartesian magic.