Color $2$-switches and neighborhood $\lambda$-balanced graphs with $k$ colors

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, $\lambda$-balanced graphs, in which any two color classes in a vertex neighborhood differ in size by at most $\lambda$. 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 $\lambda$ for which a graph admits a balanced coloring defines its $\lambda$-balance number. We prove general results about these classes and their $\lambda$-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 $\lambda \le 1$ 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.