The 1-2-3 Conjecture asserts that, for every connected graph different from K2 , its edges can be labeled with 1,2,3 so that, when coloring each vertex with the sum of its incident labels, no two adjacent vertices get the same color. This conjecture takes place in the more general context of distinguishing labelings, where the goal is to label graphs so that some pairs of their elements are distinguishable relatively to some parameter computed from the labeling. In this work, we investigate the consequences of labeling graphs as in the 1-2-3 Conjecture when it is further required to make the maximum resulting color as small as possible. In some sense, we aim at producing a number of colors that is as close as possible to the chromatic number of the graph. We first investigate the hardness of determining the minimum maximum color by a labeling for a given graph, which we show is NP-complete in the class of bipartite graphs but polynomial-time solvable in the class of graphs with bounded treewidth. We then provide bounds on the minimum maximum color that can be generated both in the general context, and for particular classes of graphs. Finally, we study how using larger labels permits to reduce the maximum color.

中文翻译：

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关于最小化 1-2-3 猜想的最大颜色

1-2-3 猜想断言，对于每个不同于 K2 的连通图，它的边可以用 1,2,3 标记，这样，当用其事件标签的总和为每个顶点着色时，没有两个相邻的顶点得到相同的颜色。这个猜想发生在区分标签的更一般的上下文中，目标是标记图，以便它们的某些元素对相对于从标签计算出的某些参数是可区分的。在这项工作中，我们研究了在 1-2-3 猜想中标记图的后果，当进一步需要使最大结果颜色尽可能小时。从某种意义上说，我们的目标是产生尽可能接近图形色数的多种颜色。我们首先研究通过给定图的标记来确定最小最大颜色的难度，我们表明它在二部图类中是 NP 完全的，但在具有有界树宽的图类中是多项式时间可解的。然后，我们提供了可以在一般上下文和特定类别的图形中生成的最小最大颜色的界限。最后，我们研究如何使用更大的标签来减少最大颜色。