论文标题
奇数$ C_4 $ -FACE-MAGIC $ M \ times n $投影网格图具有$ C_4 $ -FACE-MAGIC值$ 2MN+1 $或2MN+3 $
Odd order $C_4$-face-magic $m \times n$ projective grid graphs having $C_4$-face-magic value $2mn+1$ or $2mn+3$
论文作者
论文摘要
对于图形$ g =（v，e）$嵌入在投影平面中的$，令$ \ mathcal {f}（g）$表示$ g $的面孔。然后，$ g $如果存在Biokeve $ f：v（g）\ to \ {1，2，\ dots，| v（g）| \} $，则称为$ c_n $ -face-magic投射图。 $ S $。我们考虑$ \ Mathcal {p} _ {m，n} $表示的$ M \ times n $网格图，以自然方式嵌入了投影平面。令$ m \ geqslant 3 $和$ n \ geqslant 3 $为奇数。众所周知，$ C_4 $ -FACE-MAGIC值的$ C_4 $ -FACE-MAGIC在$ \ Mathcal {p} _ {m，n} $上是$ 2MN+1 $，$ 2MN+2 $，或者是2MN+2 $或2MN+3 $。 $ C_4 $ -FACE-MAGIC在$ \ Mathcal {p} _ {m，n} $上具有$ C_4 $ -FACE-MAGIC值$ 2MN+2 $的表征是已知的。在本文中，我们确定了$ \ \ mathcal {p} _ {m，n} $上的$ C_4 $ -FACE-MAGIC标签的类别，其中$ C_4 $ -Face-Magic Value是$ 200万美元+1 $ 1 $或200万美元+3 $。猜想这些是$ \ Mathcal {p} _ {m，n} $具有$ C_4 $ -C_4 $ -FACE-MAGIC值$ 2MN+1 $或2MN+3 $的$ C_4 $ -FACE-MAGIC标签。
For a graph $G = (V, E)$ embedded in the projective plane, let $\mathcal{F}(G)$ denote the set of faces of $G$. Then, $G$ is called a $C_n$-face-magic projective graph if there exists a bijection $f: V(G) \to \{1, 2, \dots, |V(G)|\}$ such that for any $F \in \mathcal{F}(G)$ with $F \cong C_n$, the sum of all the vertex labels around $C_n$ is a constant $S$. We consider the $m \times n$ grid graph, denoted by $\mathcal{P}_{m,n}$, embedded in the projective plane in the natural way. Let $m \geqslant 3$ and $n \geqslant 3$ be odd integers. It is known that the $C_4$-face-magic value of a $C_4$-face-magic labeling on $\mathcal{P}_{m,n}$ is either $2mn+1$, $2mn+2$, or $2mn+3$. The characterization of $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+2$ is known. In this paper, we determine a category of $C_4$-face-magic labelings on $\mathcal{P}_{m,n}$ for which the $C_4$-face-magic value is either $2mn+1$ or $2mn+3$. It is conjectured that these are the only $C_4$-face-magic labeling on $\mathcal{P}_{m,n}$ having $C_4$-face-magic value $2mn+1$ or $2mn+3$.
