图的超边幻亏数的最新研究
Recent studies on the super edge-magic deficiency of graphs
摘要：
一个图$G$称为边幻图，如果存在一个双射函数$f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$，使得对于每个$uv\in E\left( G\right) $，$f\left(u\right) + f\left(v\right) + f\left(uv\right)$都是常数.此外，如果$f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$的话，$G$据说是超级边缘魔法。进一步地，定义了图$G$的超边幻亏数$ μ_{s}\left(G\right)$为最小非负整数$n$，且$G \cup nK_{1}$是超边幻亏数，如果不存在最小非负整数$n$，则为$+ \infty$.本文引入参数$l\left(n\right)$作为$n$阶图$G$的最小尺寸，使得所有$n$阶且尺寸至少为$l\left(n\right)$的图都有$μ_{s} \left( G \right)=+\infty $，并给出了$l\left(G\right)$的上下界. Imran、Baig和Feunovucíková建立了对于具有$n\equiv 0\pmod{4}$、$ μ_{s}\left(D_{n}\right) \leq 3n/2-1$的整数$n$，其中$D_{n}$是$n$阶的循环$C_{n}$和$2$阶的完全图$K_{2}$的笛卡儿积。我们通过证明当$n \geq 4$为偶数时$ μ_{s}\left(D_{n}\right) \leq n+1$为偶数来改进这个界限。Enomoto、Lladó、Nakamigawa和Ringel提出了一个猜想：每一棵非平凡树都是超边幻树。我们提出了一种新的方法来攻击这个猜想。这种方法也可能有助于解决Graham和Sloane关于树的另一个标号猜想。
英文摘要：
A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$ such that $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant for each $uv\in E\left( G\right) $. Also, $G$ is said to be super edge-magic if $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$. Furthermore, the super edge-magic deficiency $ μ_{s}\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G \cup nK_{1}$ is super edge-magic or $+ \infty$ if there exists no such integer $n$. In this paper, we introduce the parameter $l\left(n\right)$ as the minimum size of a graph $G$ of order $n$ for which all graphs of order $n$ and size at least $l\left(n\right)$ have $μ_{s} \left( G \right)=+\infty $, and provide lower and upper bounds for $l\left(G\right)$. Imran, Baig, and Feunovucíková established that for integers $n$ with $n\equiv 0\pmod{4}$, $ μ_{s}\left(D_{n}\right) \leq 3n/2-1$, where $D_{n}$ is the cartesian product of the cycle $C_{n}$ of order $n$ and the complete graph $K_{2}$ of order $2$. We improve this bound by showing that $ μ_{s}\left(D_{n}\right) \leq n+1$ when $n \geq 4$ is even. Enomoto, Lladó, Nakamigawa, and Ringel posed the conjecture that every nontrivial tree is super edge-magic. We propose a new approach to attak this conjecture. This approach may also help to resolve another labeling conjecture on trees by Graham and Sloane.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
