arXiv 2203.004092022-03math.CO

整和图的边染色与边和染色的研究

A study on edge coloring and edge sum coloring of integral sum graphs

V. Vilfred Kamalappan, Lowell W. Beineke, L. Mary Florida, Julia K. Abraham

摘要：

Frank Harary引入了整和图的概念。一个图$G$是一个\emph{ integral sum graph}，如果它的顶点可以用不同的整数标记，使得$e = uv$是$G$的一条边当且仅当顶点$u$和$v$上的标记之和也是$G.$中的一个标记对于任何非空的整数集合$S$，令$G^+(S)$表示集合$S$上的整数和图.在$G^+(S)$中，我们定义\emph{edge-sum class}为具有相同边和数的所有边的集合，并且如果每个边和类被认为是$G^+(S)$的边颜色类，则称$G^+(S)$为\emph{edge sum color graph}。$G^+(S)$的不同边和类的数目称为它的\emph{ edge sum chromatic number}。本文的主要结果是：（i）整和图的所有边和类的集合划分它的边集;（ii）整和图$G_{0,s}$与$S_n$，$n$阶星图的边色数与边和色数相等，而$G_{r,s} = G^+([r,s])$，$r < 0 < s$，$n,s \geq 2$，$n,r,s\in\mathbb{N}$的情况则不相等.我们还得到了一个有趣的星图的整和标号。

英文摘要：

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.

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摘要：

英文摘要：

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