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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.05705 |
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| _version_ | 1866911490129264640 |
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| author | Collins, Karen L. Hook, Jonelle McBee, Cayla Trenk, Ann N. |
| author_facet | Collins, Karen L. Hook, Jonelle McBee, Cayla Trenk, Ann N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05705 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Color $2$-switches and neighborhood $λ$-balanced graphs with $k$ colors Collins, Karen L. Hook, Jonelle McBee, Cayla Trenk, Ann N. Combinatorics 05C15, 05C07 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. |
| title | Color $2$-switches and neighborhood $λ$-balanced graphs with $k$ colors |
| topic | Combinatorics 05C15, 05C07 |
| url | https://arxiv.org/abs/2603.05705 |