发表机构
HUN-REN Alfréd Rényi Institute of Mathematics; University of Pannonia; Bolyai Institute, University of Szeged; University of Johannesburg（HUN-REN 阿尔弗雷德·雷尼数学研究所; 佩奇大学; 塞格德大学博莱伊研究所; 约翰内斯堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文结合小单色分量与无孤立点两个染色性质，证明存在常数c使每个三次图可2-染色，使每个单色分量顶点数介于2和c之间，并给出非平衡版本，作为Wegner猜想工具的松弛。
AI 中文摘要
我们将两个方向相反的染色方面结合起来。一方面，可以对三次图的顶点进行2-染色，使得每个单色分量都非常小。另一方面，也可以对三次图的顶点进行2-染色，使得每个单色分量至少包含一条边（即每个单色分量中的每个顶点度至少为1）。作为解决Wegner猜想特例的预期工具，Thomassen提出了一个结合上述两个性质的猜想，这引出了crumby染色的概念。然而，事实证明存在没有这种染色的三次图。在此，我们尝试探讨对每个三次图而言，原始概念的自然松弛可能成立的情况。我们证明存在一个常数c，使得每个三次图都有一个顶点2-染色，其中每个单色分量至少包含2个顶点且至多包含c个顶点。我们还证明了一个非平衡版本，这是crumby染色的自然松弛。
英文摘要
We combine two coloring aspects that work in opposite directions. One can 2-color the vertices of a cubic graph such that each monochromatic component is very small. One can also 2-color the vertices of a cubic graph such that each monochromatic component has degree at least 1. As an intended tool for solving a special case of Wegner's conjecture, Thomassen formulated a conjecture that combined the two previous properties. This led to the concept of a crumby coloring. However it turned out that there are cubic graphs without such coloring. Here we try to see what natural relaxations of the original concept might hold for each cubic graph. We show there exists a constant $c$ such that every cubic graph has a vertex 2-coloring such that every monochromatic component has at least 2 and at most $c$ vertices. We also prove an unbalanced version, which is the natural relaxation of the crumby coloring.
