论文标题
关于图的超级边缘魔术缺陷的最新研究
Recent studies on the super edge-magic deficiency of graphs
论文作者
论文摘要
图$ g $如果存在某种射击功能$ f：v \ left（g \右） \ right \} $使得$ f \ left（u \ right） + f \ left（v \ right） + f \ left（uv \ right）$是每个$ uv \ e \ left（g \ right）$的常数。另外，如果$ f \ left（v \ left（g \ right）\ right）= \ left \ {1，2，\ ldots，\ left \ left \ vert v \ left（g \ oyt）此外，图$ g $的超级边缘魔术缺陷$μ_{s} \ left（g \ right）$被定义为最小的非负整数$ n $，而$ g \ cup g \ cup nk_ {1} $是超级边缘 - 超级 - ++\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\乐光服配米是超级边缘或$+\\\\\\\\\\\\\\\\\\决在本文中，我们将参数$ l \ left（n \右）$作为图$ g $的最小尺寸$ n $的最小尺寸，其所有图$ n $和大小至少$ l \ weft（n \右）均具有$μ_______________________________________________{s} \ left（g \ right（g \ right）=+\ iffty $，并提供$ low lows $ $ $ $ welds $ weld（$）。 Imran，Baig和Feunovucíková确定，对于Integers $ n $，带有$ n \ equiv 0 \ equiv 0 \ pmod {4} $，$μ__{s} \ left（d_ {n} \ right）\ leq 3n/2-1 $ $ k_ {2} $的订单$ 2 $。我们通过表明$μ_{s} \ left（d_ {n} \ right）\ leq n+1 $时，我们改善了这一约束。 Inomoto，Lladó，Nakamigawa和Ringel提出了这样一种猜想，即每个非平凡的树都是超级边缘魔术。我们提出了一种新的猜想方法。这种方法还可能有助于解决格雷厄姆和斯隆在树上的另一个标签猜想。
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.
