Cartesian magicness of 3-dimensional boards

Abstract

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.

On local antimagic chromatic number of cycle-related join graphs

“…When m = 1, we get P 2 ∨ O 2n = K 1,1,2n . In [15,Theorem 2.7], the authors proved that χ la (K 1,1,2n ) = 3. When n = 1, the authors in [17,Theorem 2.4] proved that χ la (P 2m ∨ O 2 ) = 3 for m ≥ 2.…”

Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number

“…. , a t ) are C 16 , C 16 (1, 3), C 16 (1,5), C 16 (1, 7), C 16 (1,3,5), C 16 (1,3,7), C 16 (1,5,7), C 16 (1,3,5,7). (1,5).…”

“…Define φ : Z 16 → Z 16 by φ(i) = 5i and ψ : Z 16 → Z 16 by ψ(i) = 3i. It is easy to check that ψ and φ induce isomorphisms from C 16 (1,3,5) to C 16 (1,3,7) and C 16 (1,5,7), respectively. However, C 16 (1, 3) ∼ = C 16 (1,7).…”

“…However, C 16 (1, 3) ∼ = C 16 (1,7). We may consider the spectra of C 16 (1,3) and C 16 (1,7). One may find the formula of the spectrum of a circulant matrix from [8].…”

“…By Theorem 2.3 the local antimagic chromatic number of each graph listed in Example 2.1 is 3. As an example to illustrate the proof of Theorem 2.3, let us consider C 16 (1,3). Now Γ 3 = (0, 3, 6, 9, 12, 15, 2, 5, 8, 11, 14, 1, 4, 7, 10, 13).…”