I am interested in (proper) vertex colorings of graphs with the following condition:
for every vertex $v$ in the graph, all colors should be present in the closed neighborhood of $v$.

Is this studied in the literature? The closest I found is $r$-dynamic $k$-coloring. For $k$-regular graphs, a ($k-1$)-dynamic $k$-coloring satisfies the above condition. But, I am also interested in a coloring that use less than $k$ colors and satisfies the above condition. Currently, I am more interested in such coloring of regular graphs.

If there is a paper dealing with ($k-1$)-dynamic $k$-colorings of $k$-regular graphs, I would like to read it (I saw a result on ($k+1$)-dynamic $k$-colorings in a paper).

Thank you.

Related Colouring Variants:-

  1. $r$-dynamic $k$-coloring (also called $r$-hued $k$-coloring)
    An $r$-dynamic $k$-coloring of $G$ is a (proper) $k$-coloring of $G$ such that every vertex $v$ in $G$ has neighbors in at least $\min\{deg(v),r\}$ different color classes.
  2. b-coloring
    A b-coloring is a (proper) coloring such that every color class $V_i$ contains a vertex $u_i$ which has neighbors in all other color classes.
    In the coloring variant I am looking for, every vertex has neighbors in all other color classes.

For a vertex $v$ of $G$, the closed neighborhood of $G$, denoted by $N[v]$, is $v$ together with its set of neighbors.
i.e., $N[v]=\{v\}\cup\{u : uv\in E(G)\}$.


2 Answers 2


I would call this a polychromatic coloring of the closed neighborhood hypergraph. I don't think this has been studied before for general graphs.

Here is a paper studying the question when edges are colored:
Béla Bollobás, David Pritchard, Thomas Rothvoß, Alex Scott: Cover-Decomposition and Polychromatic Numbers

And here is one that studies the conflict-free coloring problem for your hypergraph:
Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, Rogers Mathew: Conflict-free coloring on closed neighborhoods of bounded degree graphs

If you replace closed neighborhood with geometric range spaces, then you can find a lot more literature, most notably my own amazing papers.


It is studied in the literature. It is the coloring variant called fall coloring introduced by Dunbar et al [1].

Quote from [1] (I have made minute changes in the language):

A coloring of a graph $G=(V,E)$ is a partition $\Pi=\{V_1,V_2,\dots,V_k\}$ of the vertices of G into independent sets $V_i$, or color classes. A vertex $v\in V_i$ is called colorful if it is adjacent to at least one vertex in every color class $V_j$, $j\neq i$. A fall coloring is a coloring in which every vertex is colorful. If a graph G has a fall coloring, the fall chromatic number of G is the minimum number of colors for a fall coloring of G.

Interestingly, the fall chromatic number of $G$ is equal to the maximum number of independent dominating sets into which the vertex set of $G$ can be partitioned (the latter is called the idomatic number of $G$). Note that an independent dominating set of $G$ is precisely a maximal independent set of $G$.

url for list of papers that cited [1] (according to google scholar): https://scholar.google.com/scholar?cites=6893367022071411040&as_sdt=2005&sciodt=0,5&hl=en

[1] Dunbar, J. E.; Hedetniemi, S. M.; Hedetniemi, S. T.; Jacobs, D. P.; Knisely, J.; Laskar, R. C.; Rall, D. F., Fall colorings of graphs, J. Comb. Math. Comb. Comput. 33, 257-273 (2000). ZBL0962.05020.

  • $\begingroup$ Another name used in the literature is frozen coloring. $\endgroup$ Commented Dec 27, 2023 at 11:52

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