The chromatic number, which refers to the minimum number of colours required to colour the vertices of graphs properly, is one of the most central notions of the graph chromatic theory. Several of its aspects of interest have been investigated in the literature, including variants for modifications of proper colourings. These variants include, notably, the achromatic number of graphs, which is the maximum number of colours required to colour the vertices of graphs properly so that each possible combination of distinct colours is assigned along some edge. The behaviours of this parameter have led to many investigations of interest, bringing to light both similarities and discrepancies with the chromatic number. This work takes place in a recent trend aiming at extending the chromatic theory of graphs to the realm of signed graphs, and, in particular, at investigating how classic results adapt to the signed context. Most of the works done in that line to date are with respect to two main generalisations of proper colourings of signed graphs, attributed to Zaslavsky and Guenin. Generalising the achromatic number to signed graphs was initiated recently by Lajou, his investigations being related to Guenin's colourings. We here pursue this line of research, but with taking Zaslavsky's colourings as our notion of proper colourings. We study the general behaviour of our resulting variant of the achromatic number, mainly by investigating how known results on the classic achromatic number generalise to our context. Our results cover, notably, bounds, standard operations on graphs, and complexity aspects.
翻译:染色体数是指正确显示图表顶部所需的最低颜色数量, 指正确显示图表顶部所需的最起码颜色数量, 是图彩色理论中最核心的概念之一。 文献中已经调查了其中几个值得关注的方面, 包括修改适当颜色的变异。 这些变异包括, 特别是, 图表的色色数, 也就是正确显示图表顶端所需的最大颜色数量, 以便在某些边缘排列每个可能的不同颜色组合。 这个参数的行为引出了许多感兴趣的调查, 揭示了与染色体数字的相似性和差异。 这项工作是在近期的趋势中进行的, 目的是将图表的染色学理论扩展至已签名的颜色范围, 特别是研究经典结果如何适应已签名的颜色。 该行迄今为止完成的大部分工作涉及两个关于已签名的图表的正确颜色数字的简单化分析, 归因于Zaslavsky 和 Guenin 。 将我们标定的颜色操作的直系背景数概括到签名的颜色分析结果, 最近由Lajjoch 开始, 我们的颜色直系开始, 与我们标定的颜色的颜色的常规研究结果相关。