Proper (Strong) Rainbow Connection and Proper (Strong) Rainbow Vertex Connection of Some Special Graphs

J. Interconnect. Networks Pub Date : 2023-01-20 DOI:10.1142/s0219265922500062

Yingbin Ma, Yanfeng Xue, Xiaoxue Zhang

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引用次数: 1

Abstract

The proper rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is rainbow vertex connected. The proper strong rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is strong rainbow vertex connected. These two concepts are inspired by the concept of proper (strong) rainbow connection number of graphs. In this paper, we first determine the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], pencil graphs, circular ladders or Möbius ladders. Secondly, we obtain the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], paths, cycles, wheels, complete multipartite graphs, pencil graphs, circular ladders and Möbius ladders. Finally, we characterize all the connected graphs [Formula: see text] with [Formula: see text] and [Formula: see text].

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某些特殊图的固有(强)彩虹连接和固有(强)彩虹顶点连接

【公式:见文】的彩虹顶点连接数，用【公式:见文】表示，是为【公式:见文】的顶点正确上色，使【公式:见文】的彩虹顶点连接所需的最小颜色数。【公式:见文】的适当强彩虹顶点连接数，用【公式:见文】表示，是为【公式:见文】的顶点适当上色，使【公式:见文】为强彩虹顶点连接所需的最小颜色数。这两个概念的灵感来自于图的固有(强)彩虹连接数的概念。在本文中，我们首先确定了一些特殊图的[公式:见文]和[公式:见文]的值，例如所有的三次有序图[公式:见文]，铅笔图，圆形阶梯或Möbius阶梯。其次，我们得到了一些特殊图的[公式:见文]和[公式:见文]的值，如所有有序的三次图[公式:见文]、路径、循环、轮子、完全多部图、铅笔图、圆形阶梯和Möbius阶梯。最后，我们用[公式:见文本]和[公式:见文本]来描述所有的连通图[公式:见文本]。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

J. Interconnect. Networks

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