Cartesian magicness of 3-dimensional boards
Downloads
DOI:
https://doi.org/10.26637/MJM0803/0077Abstract
A $(p, q, r)$-board that has $p q+p r+q r$ 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 $f: S \rightarrow[1, p q+p r+q r]$. Firstly, for $1 \leq i \leq p$, 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 $1 \leq j \leq q$, 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 $1 \leq k \leq r$, 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 $\left\{x_i: 1 \leq i \leq p\right\}=\left\{c_1\right\}$ for some constant $c_1,\left\{y_j: 1 \leq j \leq q\right\}=\left\{c_2\right\}$ for some constant $c_2$, and $\left\{z_k: 1 \leq k \leq r\right\}=\left\{c_3\right\}$ 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 magicMathematics Subject Classification:
Mathematics- Pages: 1175-1185
- Date Published: 01-07-2020
- Vol. 8 No. 03 (2020): Malaya Journal of Matematik (MJM)
S. Arumugam, K. Premalatha, M. Bacǎ and A. Semaničová-Feňovčíková, Local antimagic vertex coloring of a graph, Graphs and Combin., 33(2017), 275-285.
J. Bensmail, M. Senhaji and K. Szabo Lyngsie, On a combination of the 1-2-3 Conjecture and the Antimagic Labelling Conjecture, Discrete Math. Theoret. Comput. Sci., 19(1)(2017) #22.
E.S. Chai, A. Das and C.K. Midha, Construction of magic rectangles of odd order, Australas. J. Combin., 55(2013), 131-144.
J.P. De Los Reyes, A. Das and C.K. Midha, A matrix approach to construct magic rectangles of even order, Australas. J. Combin., 40(2008), 293-300.
G.C. Lau, W.C. Shiu and H.K. Ng, Affirmative solutions on local antimagic chromatic number, Graphs and Combinatorics, (Online 2020), https://doi.org/10.1007/s00373020-02197-2
G.C. Lau, W.C. Shiu and H.K. Ng, On local antimagic chromatic number of cycle-related join graphs, Discuss. Math. Graph Theory, (Online 2018), https://doi.org/10.7151/dmgt.2177
G.C. Lau, W.C. Shiu and H.K. Ng, On local antimagic chromatic number of graphs with cut-vertices, (2018) arXiv: 1805.04801
J.P.N. Phillips, The use of magic squares for balancing and assessing order effects in some analysis of variance designs, Appl. Statist., 13(1964), 67-73.
J.P.N. Phillips, A simple method of constructing certain magic rectangles of even order, Math. Gazette, 52(1968a), $9-12$.
J.P.N Phillips, Methods of constructing one-way and factorial designs balanced for tread, Appl. Statist., 17(1968b), $162-170$.
W.C. Shiu, P.C.B. Lam and S-M. Lee, Edge-magicness of the composition of a cycle with a null graph, Congr. Numer., 132(1998), 9-18.
W.C. Shiu, P.C.B. Lam and S-M. Lee, On a construction of supermagic graphs, $J C M C C, 42(2002), 147-160$.
B.M. Stewart, Magic graphs, Canadian Journal of Mathematics, 18(1966), 1031-1059.
B.M. Stewart, Supermagic Complete Graphs, Canadian Journal of Mathematics, 19(1967), 427-438.
- NA
Similar Articles
- Jaydeep Parejiya, Patat Sarman, Pravin Vadhel, Planarity of a unit graph: Part-I local case , Malaya Journal of Matematik: Vol. 8 No. 03 (2020): Malaya Journal of Matematik (MJM)
- Jaydeep Parejiya, Patat Sarman, Pravin Vadhel, Planarity of a unit graph part -III \(|Max (R)| \geq 3\) case , Malaya Journal of Matematik: Vol. 8 No. 04 (2020): Malaya Journal of Matematik (MJM)
- Jaydeep Parejiya, Pravin Vadhel, Patat Sarman, Planarity of a unit graph: Part -II \(|Max(R)|= 2\) case , Malaya Journal of Matematik: Vol. 8 No. 03 (2020): Malaya Journal of Matematik (MJM)
You may also start an advanced similarity search for this article.
Metrics
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 MJM
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.