Cube-magic labelings of grids

TL;DR

This paper proves that vertices and edges of multi-dimensional grid graphs can be labeled to ensure all subcubes have the same sum, establishing a supermagic property for these grids.

Contribution

It introduces a new labeling method for grid graphs that guarantees uniform sums over all subcubes, extending supermagic labelings to higher-dimensional grids.

Findings

Vertices and edges of grid graphs can be labeled to achieve supermagic sums.

All subgraphs isomorphic to a d-cube have the same total label sum.

The method applies to any dimension d ≥ 2.

Abstract

We show that the vertices and edges of a $d$-dimensional grid graph $G=(V,E)$ ($d\geqslant 2$) can be labeled with the integers from $\{1,\ldots,\lvert V\rvert\}$ and $\{1,\ldots,\lvert E\rvert\}$, respectively, in such a way that for every subgraph $H$ isomorphic to a $d$-cube the sum of all the labels of $H$ is the same. As a consequence, for every $d\geqslant 2$, every $d$-dimensional grid graph is $Q_d$-supermagic where $Q_d$ is the $d$-cube.

Equations

$$\sum _{v\in V(H^{\prime })}F(v)+\sum _{e\in E(H^{\prime })}F(e)=c$$

$$E=\left\{\{\mbox{\boldmath$x$},\mbox{\boldmath$y$}\}\ :\ \mbox{\boldmath$x$},\,\mbox{\boldmath$y$}\in V\text{ and }\sum_{i=1}^{d}\lvert x_{i}-y_{i}\rvert=1\right\}.$$

$$\sum _{v\in V(H^{\prime })}f(v)=c$$

$$\sum _{e\in E(H^{\prime })}f(e)=c^{\prime }$$

$$[P]=\{\begin{array}{ll}1& \text{if P is true,}\\ 0& \text{if P is false.}\end{array}$$

$$f(i,j)=\{\begin{array}{ll}(i-1)n_2+j& \text{if i is odd and j is odd},\\ (i-1)n_2+(n_2+1-j)& \text{if i is even and j is even},\\ (n_1-i)n_2+j+[2\mid n_1\wedge 2\nmid n_2]& \text{if i is odd and j is even},\\ (n_1-i)n_2+(n_2+1-j+[2\mid n_1\wedge 2\nmid n_2])& \text{if i is even and j is odd},\end{array}$$

$$g((i,j),(i,j+1))$$

$$g((i,j),(i+1,j))$$

$$c(f)=\{\begin{array}{ll}2(n_1{n}_2+1)& \text{if n1 is odd or n2 is even},\\ 2(n_1{n}_2+2)& \text{if n1 is even and n2 is odd}.\end{array}$$

$$f(i,j)$$

$$=(i-1)n_2+j+(n_1-i)n_2+(j+1)+[2\mid n_1\wedge 2\nmid n_2]+(n_1-(i+1))n_2$$

$$+n_2+1-(j+1)+[2\mid n_1\wedge 2\nmid n_2]+((i+1)-1)n_2+n_2+1-(j+1)$$

$$=(2n_1-2)n_2+2n+2+2[2\mid n_1\wedge 2\nmid n_2]$$

$$=2(n_1{n}_2+1+[2\mid n_1\wedge 2\nmid n_2]).\text{qed}$$

$$g((i,j)$$

$$=(i-1)(2n_2-1)+j+(n_1-i)(2n_2-1)+1-j+((i+1)-1)(2n_2-1)+j$$

$$+(n_1-i)(2n_2-1)+1-(j+1)$$

$$=(2n_1-1)(2n_2-1)+1.\text{qed}$$

$$c(F)=10n_1{n}_2-2n_1-2n_2+\{\begin{array}{ll}4& \text{if n1 is odd or n2 is even},\\ 6& \text{if n1 is even and n2 is odd}.\end{array}$$

$$f(x_1,\dots ,x_d)=\{\begin{array}{ll}\tilde{f}(x_1,\dots ,x_{d-1})+(x_d-1)\prod _{i=1}^{d-1}{n}_i& \text{if }{x}_1+\cdots +x_{d-1}\text{ is even},\\ \tilde{f}(x_1,\dots ,x_{d-1})+(n_d-x_d)\prod _{i=1}^{d-1}{n}_i& \text{if }{x}_1+\cdots +x_{d-1}\text{ is odd}.\end{array}$$

$$c(f)=2S+2^{d-1}(n_d-1)\prod _{i=1}^{d-1}{n}_i.$$

$$27$$

$$42$$

$$V(H)=\left\{(x_1+{\epsilon }_1,\dots ,x_{d-1}+{\epsilon }_{d-1},i):({\epsilon }_1,\dots ,{\epsilon }_{d-1})\in \{0,1{\}}^{d-1}\right\}.$$

$$\sum_{v\in V(H)}f(v)=\sum_{\mbox{\boldmath$\varepsilon$}\in\{0,1\}^{d-1}}\tilde{f}(x_{1}+\varepsilon_{1},\ldots,x_{d-1}+\varepsilon_{d-1})+2^{d-2}(i-1)\prod^{d-1}_{i=1}n_{i}+2^{d-2}(n_{d}-i)\prod^{d-1}_{i=1}n_{i}\\ =S+2^{d-2}\prod^{d-1}_{i=1}n_{i}(i-1+n_{d}-i)=S+2^{d-2}(n_{d}-1)\prod^{d-1}_{i=1}n_{i}.$$

$$V(H)=\{(x_1+{\epsilon }_1,\dots ,x_{d-1}+{\epsilon }_{d-1},x_d+{\epsilon }_d):({\epsilon }_1,\dots ,{\epsilon }_{d-1},{\epsilon }_d)\in \{0,1{\}}^d\}$$

$$\sum _{v\in V(H)}f(v)=2\left(S+2^{d-2}(n_d-1)\prod _{i=1}^{d-1}{n}_i\right).\text{qed}$$

$$g(e)=\tilde{f}(x_1,\dots ,x_{d-1})+n_{d}M+\{\begin{array}{ll}(x_d-1)N& \text{ if }{x}_1+\cdots +x_{d-1}\text{ is odd},\\ (n_d-1-x_d)N& \text{ if }{x}_1+\cdots +x_{d-1}\text{ is even}.\end{array}$$

$$g(e)=\tilde{g}(\{(x_1,\dots ,x_{d-1}),(y_1,\dots ,y_{d-1})\})+\{\begin{array}{ll}(x_d-1)M& \text{ if }i\text{ is odd},\\ (n_d-x_d)M& \text{ if }i\text{ is even}.\end{array}$$

$$g(e)=\tilde{g}(\{(x_1,\dots ,x_{d-1}),(y_1,\dots ,y_{d-1})\})+\{\begin{array}{ll}(x_d-1)M& \text{ if }{\sum }_{i=1}^{d-2}{x}_i\text{ is odd},\\ (n_d-x_d)M& \text{ if }{\sum }_{i=1}^{d-2}{x}_i\text{ is even}.\end{array}$$

$$c^{\prime }(g)=S+2S^{\prime }+2^{d-2}(n_d-2)N+2^{d-2}\left(2n_d+(d-1)(n_d-1)\right)M.$$

$$78$$

$$15$$

$$37$$

Full text

1 Introduction

The graphs considered in this paper are finite, undirected and simple. For a graph $G$, we denote its vertex set by $V(G)$ and its edge set by $E(G)$. A graph labeling, as introduced in [8], is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Over the years, a large variety of different types of graph labelings have been studied, see [2] for an extensive survey.

For a graph $H$, we say that a graph $G$ admits an $H$-covering if every edge of $G$ belongs to at least one subgraph of $G$ which is isomorphic to $H$. A graph $G=(V,E)$ which admits an $H$-covering is called $H$-magic if there exists a bijection $F:V\cup E\to\{1,2,\ldots,\lvert V\rvert+\lvert E\rvert\}$ and a constant $c=c(F)$, which we call the $H$-magic sum of $F$, such that

[TABLE]

for every subgraph $H^{\prime}\subseteq G$ with $H^{\prime}\cong H$. If in addition $F(V)=\{1,\ldots,\lvert V\rvert\}$ then we say that the graph $G$ is $H$-supermagic. The case where $H$ is a single edge was studied in [1], and the general concept for arbitrary graphs $H$ was introduced in [3]. Since then $H$-magic and $H$-supermagic labelings have been studied for a variety of graphs $H$ ([4, 5, 6, 7, 9]).

In this paper we show that for an integer $d\geqslant 2$, a $d$-dimensional grid graph $G$ is $Q_{d}$-supermagic where $Q_{d}$ denotes the $d$-cube. For $d=2$, the $2$-cube is the same as a 4-cycle $C_{4}$ and our result is a consequence of Theorem 1 in [4] which gives sufficient conditions for the cartesian product of a graph and a path to be $C_{4}$-supermagic.

The structure of the paper is as follows. In Section 2 we fix some notation and state our main result. Section 3 contains the proof which is by induction on the dimension $d$, where the base case $d=2$ is contained in Section 3.1 and the induction step in Section 3.2.

2 Notation and main result

For integers $k\leqslant\ell$ we denote the sets $\{1,2,\ldots,k\}$ and $\{k,k+1,\ldots,\ell\}$ by $[k]$ and $[k,\ell]$, respectively. For integers $d\geqslant 2$ and $n_{1}\geqslant n_{2}\geqslant\cdots\geqslant n_{d}\geqslant 2$, let $\textsc{Grid}(d_{1},\ldots,n_{d})$ denote the $n_{1}\times\cdots\times n_{d}$-grid graph, i.e., the cartesian product of $d$ paths of lengths $n_{1}$,…, $n_{d}$. In other words, the vertex set of $G(d_{1},\ldots,n_{d})$ is $V=[n_{1}]\times\cdots\times[n_{d}]$ and edge set

[TABLE]

The graph $\textsc{Grid}(2,2,\ldots,2)$ is called the $d$-cube and will be denoted by $Q_{d}$.

To simplify the presentation of our proof we will label the vertices and the edges separately and then combine the labelings to obtain the $Q_{d}$-supermagic labeling. A vertex labeling $f:V\to\{1,2,\ldots,\lvert V\rvert\}$ for a graph $G=(V,E)$ is called $H$-magic if there exists a constant $c=c(f)$, called the $H$-magic sum of $f$ such that

[TABLE]

for every subgraph $H^{\prime}\subseteq G$ with $H^{\prime}\cong H$. Similarly, an edge labeling $g:E\to\{1,2,\ldots,\lvert E\rvert\}$ for a graph $G=(V,E)$ is called $H$-magic if there exists a constant $c^{\prime}=c^{\prime}(g)$, called the $H$-magic sum of $f$ such that

[TABLE]

for every subgraph $H^{\prime}\subseteq G$ with $H^{\prime}\cong H$. An $H$-magic vertex labeling $f$ and an $H$-magic edge labeling $g$ with $H$-magic sums $c=c(f)$ and $c^{\prime}=c^{\prime}(g)$ can be combined to obtain an $H$-supermagic labeling $F$ with $H$-supermagic sum $c+\lvert E(H)\rvert\lvert V(G)\rvert$ by setting $F(v)=f(v)$ for all $v\in V(G)$ and $F(e)=g(e)+\lvert V(G)\rvert$ for all $e\in E(G)$.

Theorem 1**.**

Let $d\geqslant 2$ and $n_{1}\geqslant n_{2}\geqslant\cdots\geqslant n_{d}\geqslant 2$ be positive integers, and let $G=\textsc{Grid}(n_{1},\ldots,n_{d})$. Then $G$ admits a $Q_{d}$-magic vertex labeling $f$ and a $Q_{d}$-magic edge labeling $g$.

Based on the observation about combining $H$-magic vertex and edge labelings we obtain the following corollary.

Corollary 1**.**

Let $d\geqslant 2$ and $n_{1}\geqslant n_{2}\geqslant\cdots\geqslant n_{d}\geqslant 2$ be positive integers. Then $\textsc{Grid}(n_{1},\ldots,n_{d})$ is $Q_{d}$-supermagic.

3 Proof of the main result

We proceed by induction on $d$. In Section 3.1 we treat the base case $d=2$, and present explicit vertex and edge labelings for grid graphs $\textsc{grid}(n_{1},n_{2})$. In Section 3.2 we assume $d\geqslant 3$, and we describe how labelings $f$ and $g$ for $\textsc{Grid}(n_{1},\ldots,n_{d})$ can be constructed from the labelings $\tilde{f}$ and $\tilde{g}$ for $\textsc{Grid}(n_{1},\ldots,n_{d-1})$.

3.1 The base case $d=2$

In order to describe the labeling in a compact way we use $[P]$ to denote the indicator function for a statement $P$, i.e.,

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We define a vertex labeling $f:[n_{1}]\times[n_{2}]\to[n_{1}n_{2}]$ by

[TABLE]

and an edge labeling $g:E\to[2n_{1}n_{2}-n_{1}-n_{2}]$ by

[TABLE]

Example 1**.**

The construction is illustrated in Figure 1 for the graph $\textsc{Grid}(5,3)$.

Lemma 1**.**

The function $f$ defined by (1) is a $Q_{2}$-magic vertex labeling for $\textsc{Grid}(n_{1},n_{2})$ with $Q_{2}$-magic sum

[TABLE]

Proof.

For every $(i,j)\in[n_{1}-1]\times[n_{2}-1]$ we have

[TABLE]

Lemma 2**.**

The function $g$ defined by (2) and (3) is a $Q_{2}$-magic edge labeling for $\textsc{Grid}(n_{1},n_{2})$ with $Q_{2}$-magic sum $c(g)=(2n_{1}-1)(2n_{2}-1)+1$.

Proof.

For every $(i,j)\in[n_{1}-1]\times[n_{2}-1]$, we have

[TABLE]

Combining these two labelings as described in Section 2 we obtain a $Q_{2}$-supermagic labeling $F$ with $Q_{2}$-supermagic sum

[TABLE]

3.2 The induction step

We now assume $d\geqslant 3$. By induction, there exist $Q_{d-1}$-magic labelings $\tilde{f}$ and $\tilde{g}$ with $Q_{d-1}$-magic sums $S=c(f)$ and $S^{\prime}=c^{\prime}(g)$ for $\textsc{Grid}(n_{1},\ldots,n_{d-1})$. We define $f:[n_{1}]\times\cdots\times[n_{d}]\to\left\{1,\ldots,\prod^{d}_{i=1}n_{i}\right\}$ by

[TABLE]

Example 2**.**

The vertex labeling (4) is illustrated in Figure 2 for the graph $\textsc{Grid}(5,3,3)$ where $\tilde{f}$ is the vertex labeling for $\textsc{Grid}(5,3)$ presented in Example 1.

Lemma 3**.**

The function $f$ defined by (4) a $Q_{d}$-magic vertex labeling with $Q_{d}$-magic sum

[TABLE]

Before proving the lemma we illustrate the basic idea using the $Q_{3}$-subgraph of $\textsc{Grid}(5,3,3)$ shown in Figure 3.

This subgraph corresponds to the square labeled 12–7–8–5 in Figure 1, and by (4) the vertex labels for the top and the bottom square of the cube in Figure 3 are

[TABLE]

and therefore the sum of the vertex labels is $2(12+7+8+5+4\times 15)=2S+8\times 15$, where $S$ is the magic sum of the vertex labeling in Figure 1.

Proof of Lemma 3.

Fix $i\in[n_{d}]$ and $(x_{1},\ldots,x_{d-1})\in[n_{1}-1]\times\cdots\times[n_{d-1}-1]$. Let $H\cong Q_{d-1}$ be the subgraph of $\textsc{Grid}(n_{1},\ldots,n_{d})$ induced by

[TABLE]

Using the fact that exactly half of the $2^{d-1}$ vertices of $H$ have even coordinate sum, we obtain from (4),

[TABLE]

A subgraph $H$ of $\textsc{Grid}(n_{1},\ldots,n_{d})$ is isomorphic to $Q_{d}$ if and only if its vertex set is

[TABLE]

for some $(x_{1},\ldots,x_{d})\in[n_{1}-1]\times\cdots\times[n_{d}-1]$. Using (5), this implies

[TABLE]

We define an edge labeling $g:E\to\{1,\ldots,\lvert E\rvert\}$ for $G=\textsc{Grid}(n_{1},\ldots,n_{d})$ as follows. Let $N=\lvert V(\textsc{Grid}(n_{1},\ldots,n_{d-1}))\rvert$ and $M=\lvert E(\textsc{Grid}(n_{1},\ldots,n_{d-1}))\rvert$. For an edge e=\{\mbox{\boldmathx},\mbox{\boldmathy}\} with $y_{d}=x_{d}+1$, we set

[TABLE]

For the remaining edges we distinguish two cases.

Case 1.

$d$ is odd. For an edge e=\{\mbox{\boldmathx},\mbox{\boldmathy}\} with $y_{i}=x_{i}+1$, $i\leqslant d-1$, we set

[TABLE]

Case 2.

$d$ is even. For an edge e=\{\mbox{\boldmathx},\mbox{\boldmathy}\} with $y_{i}=x_{i}+1$, $i\leqslant d-2$, we set

[TABLE]

For an edge e=\{\mbox{\boldmathx},\mbox{\boldmathy}\} with $y_{d-1}=x_{d-1}+1$, we set

[TABLE]

Example 3**.**

The edge labeling given by (6) to (9) is illustrated in Figure 4 for the graph $\textsc{Grid}(5,3,3)$. The underlying labelings for $\tilde{f}$ and $\tilde{g}$ for $\textsc{Grid}(5,3)$ are the labelings from Example 1.

Lemma 4**.**

The function $g$ defined by (6) to (9) is a $Q_{d}$-magic edge labeling with $Q_{d}$-magic sum

[TABLE]

Again, we illustrate the basic idea before going into the formal proof. Figure 6 shows the edge labels for the same subgraph of $\textsc{Grid}(5,3,3)$ as in the illustration for Lemma 3 (see Figure 3).

The top and bottom squares of the cube in Figure 6 correspond to the square labeled 12–7–8–5 in Figure 6. In this example we have $N=15$ and $M=22$, and according to (6), the labels of the vertical edges are

[TABLE]

so the sum of the labels of the vertical edges is $S+12\times 22+2\times 15$, where $S=12+7+8+5$ is the vertex-magic sum in Figure 6. According to (7), the labels of the edges in the top and the bottom square are

[TABLE]

and therefore the sum of these edge labels is $2(15+11+14+6+4\times 22)=2S^{\prime}+8\times 22$, where $S^{\prime}=15+11+14+6$ is the edge-magic sum in Figure 6. Adding all edge labels we obtain $S+2S^{\prime}+2\times 15+20\times 22$ as claimed in the lemma.

Proof of Lemma 4.

Fix a subgraph $H\subseteq\textsc{Grid}(n_{1},\ldots,n_{d})$ with $H\cong Q_{d}$. Its vertex set is

[TABLE]

for some $(x_{1},\ldots,x_{d})\in[n_{1}-1]\times\cdots\times[n_{d}-1]$. We partition the vertex set as $V(H)=V_{0}(H)\cup V_{1}(H)$ and the edge set as $E(H)=E_{0}(H)\cup E_{1}(H)\cup E_{2}(H)$ where

[TABLE]

Note that $\lvert V_{0}(H)\rvert=\lvert V_{1}(H)\rvert=\lvert E_{2}(H)\rvert=\lvert V(Q_{d-1})\rvert=2^{d-1}$ and $\lvert E_{0}(H)\rvert=\lvert E_{1}(H)\rvert=\lvert E(Q_{d-1})\rvert=(d-1)2^{d-2}$. Using the fact that $y_{1}+\cdots+y_{d-1}$ is even for exactly half of the edges \{\mbox{\boldmathy},\,\mbox{\boldmathz}\}\in E_{2}(H) and that the subgraph of $G$ induced by $V_{0}(H)$ is isomorphic to $Q_{d-1}$, we obtain from (6),

[TABLE]

For the edges in $E_{0}(H)\cup E_{1}(H)$ we use the fact that each of the sets $V_{\varepsilon}(H)$ induces a subgraph isomorphic to $Q_{d-1}$. In addition, if $d$ is odd then exactly half of the indices $i\in\{1,\ldots,d-1\}$ are even, and if $d$ is even then

• •

exactly half of the indices $i\in\{1,\ldots,d-2\}$ are even, and

• •

for exactly half of the vertices $(y_{1},\ldots,y_{d-2},x_{d-1},x_{d}+\varepsilon)$ the sum $y_{1}+\cdots y_{d-2}$ is even.

In both cases we conclude that for exactly half of the edges in $E_{\varepsilon}(H)$ the term added to $\tilde{g}(\{(x_{1},\ldots,x_{d-1}),(y_{1},\ldots,y_{d-1})\})$ in (6) to (9) is $(x_{d}-1)M$, and for the other half it is $(n_{d}-x_{d})M$. This implies

[TABLE]

Combining (10) and (11), the function $g$ is a $Q_{d}$-magic labeling with $Q_{d}$-magic sum

[TABLE]