颜色2切换和具有k种颜色的邻域λ-平衡图
Color $2$-switches and neighborhood $λ$-balanced graphs with $k$ colors
AI总结：
本文研究了颜色2切换和k种颜色的λ-平衡图，提出新的图类定义并证明了相关平衡数结果。
摘要：
本文研究了具有顶点颜色分布约束的图的着色问题。我们引入了颜色2切换和颜色度数矩阵。k-着色图的颜色度数矩阵是度序列的类比，而颜色2切换提供了一种将k-着色图转换为另一种这样的图的方法，同时保持每个顶点的颜色和每个顶点邻域中的颜色多重集不变。我们证明，如果两个k-着色图具有相同的颜色度数矩阵，则当且仅当可以通过一系列颜色2切换得到对方。在相关工作中，我们通过允许k种颜色（而不是两种）和对每个邻域中各颜色顶点数量的更多灵活性，扩展了邻域平衡着色。我们引入了三种k-着色的λ-平衡图类别，其中任何两个颜色类别在顶点邻域中的大小差异最多为λ。这些类别根据是否对开邻域N(v)、闭邻域N[v]或允许每个顶点变化而区分。对于每个类别，允许平衡着色的最小λ定义了其λ-平衡数。我们证明了这些类别的通用结果及其λ-平衡数。对于k=2，我们引入了第四个类别，即奇偶平衡图，其中在偶度顶点的开邻域中每种颜色的顶点数量相等，在奇度顶点的闭邻域中每种颜色的顶点数量相等。此外，我们还关注k=2且λ≤1的重要情况，并引入了红蓝移除技术。我们提供了这四个类别之间的分离示例，并为路径、环、轮图、树、毛毛虫图和完全多部图证明了平衡数结果，并为毛毛虫图提供了计数结果。
英文摘要：
This paper examines vertex colorings of graphs with constraints on the distribution of colors in vertex neighborhoods. We introduce color 2-switches and color degree matrices. The color degree matrix of a $k$-colored graph is an analog of the degree sequence, while a color 2-switch provides a way to transform a $k$-colored graph to another such graph while maintaining the color of each vertex and the multiset of colors in each vertex neighborhood. We prove that two $k$-colored graphs have the same color degree matrix if and only if one can be obtained from the other by a sequence of color 2-switches. In related work, we generalize neighborhood balanced colorings by allowing for $k$ colors (instead of two) and more flexibility on the number of vertices of each color in a neighborhood. We introduce three classes of $k$-colored, $λ$-balanced graphs, in which any two color classes in a vertex neighborhood differ in size by at most $λ$. These classes are distinguished by whether the balancing condition is imposed on the open neighborhood $N(v)$, the closed neighborhood $N[v]$, or allowed to vary by vertex. For each class, the minimum $λ$ for which a graph admits a balanced coloring defines its $λ$-balance number. We prove general results about these classes and their $λ$-balance numbers. For $k = 2$, we introduce a fourth class, parity balanced graphs, in which the number of vertices of each color are equal in open neighborhoods for even-degree vertices and in closed neighborhoods for odd-degree vertices. Additionally, we focus on the important case where $k=2$ and $λ\le 1$ and introduce the technique of red-blue removals. We provide separating examples between these four classes and prove balance number results for paths, cycles, wheels, trees, caterpillars, and complete multipartite graphs, and a counting result for caterpillars.
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