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

Abstract

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.

Figures (10)

• Figure 1: A 2-switch in which edges $v_2v_5$ and $v_3v_4$ are replaced by $v_2v_4$ and $v_3v_5$. This is not a color $2$-switch for the first two graphs, but is a color $2$-switch transforming $G'$ into $H'$.
• Figure 2: The color degree matrices for the graphs in Figure \ref{['house-fig']} with color 1 (red, R) and color 2 (blue, B).
• Figure 3: The trees $T_1$ and $T_2$ where the ordered pair $(i,j)$ next to a vertex $v$ indicates that $v$ has $i$ red neighbors and $j$ blue neighbors.
• Figure 4: Graphs $G$ and $G'$ and $3$-colorings of them for which $D(G) = D(G')$.
• Figure 5: The graphs $K_2 \Box C_5$ and $K_4+K_3\Box K_2$ with $2$-colorings and no possible transformation via color 2-switches.

Theorems & Definitions (75)

• Definition 1.1
• Remark 1.2
• Example 1.3
• Definition 2.1
• Remark 2.2
• Example 2.3
• Theorem 2.4
• proof
• Example 2.5
• Example 2.6