论文标题
一项关于Edge着色和整体总和图的边缘总和的研究
A study on edge coloring and edge sum coloring of integral sum graphs
论文作者
论文摘要
弗兰克·哈里（Frank Harary）介绍了整体总和的概念。图$ g $是\ emph {Integral sum graph}，如果可以用不同的整数标记其顶点，以便$ e = uv $是$ g $的边缘，并且仅当$ G. $g。$ g. $ g. $ g. $ g. $ g. $ g. $ g. $ s y $ $ $ $ g $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ g^+（s）。在$ g^+（s）$中，我们将一个\ emph {edge-sum class}定义为所有边缘的集合，每个边缘都具有相同的边缘总数，并调用$ g^+（s）$ an \ emph {edge sum color图}，如果每个edge-sum类都被视为$ g^+（s）$的边缘颜色类。 $ g^+（s）$的不同边缘和类别的数量称为其\ emph {edge sum chalomatic number}。本文的主要结果是（i）积分总和图分区的所有边缘和类别的集合； （ii）Edge颜色数和边缘总和的整体总和图相等$ n，r，s \ in \ mathbb {n} $。我们还获得了恒星图的有趣的积分标记。
Frank Harary introduced the concept of integral sum graph. A graph $G$ is an \emph{ integral sum graph} if its vertices can be labeled with distinct integers so that $e = uv$ is an edge of $G$ if and only if the sum of the labels on vertices $u$ and $v$ is also a label in $G.$ For any non-empty set of integers $S$, let $G^+(S)$ denote the integral sum graph on the set $S$. In $G^+(S)$, we define an \emph{edge-sum class} as the set of all edges each with same edge sum number and call $G^+(S)$ an \emph{edge sum color graph} if each edge-sum class is considered as an edge color class of $G^+(S)$. The number of distinct edge-sum classes of $G^+(S)$ is called its \emph{ edge sum chromatic number}. The main results of this paper are (i) the set of all edge-sum classes of an integral sum graph partitions its edge set; (ii) the edge chromatic number and the edge sum chromatic number are equal for the integral sum graphs $G_{0,s}$ and $S_n$, Star graph of order $n$, whereas it is not in the case of $G_{r,s} = G^+([r,s])$, $r < 0 < s$, $n,s \geq 2$, $n,r,s\in\mathbb{N}$. We also obtain an interesting integral sum labeling of Star graphs.
