A Note on Distance-Fall Colorings
Wayne Goddard, Sonwabile Mafunda

TL;DR
This paper introduces a new type of graph coloring called distance-$k$ fall coloring, proves its existence for certain classes of graphs, and confirms a conjecture about the size of distance-$d$-dominating sets in trees.
Contribution
It establishes the existence of distance-$k$ fall colorings for connected 3-colorable graphs and trees, confirming an old conjecture about dominating sets in trees.
Findings
Connected 3-colorable graphs have a distance-2 fall 3-coloring.
Trees of order at least $k$ have a $k$-coloring with vertices within distance $k-1$ of every color.
Proves the conjecture that trees have small independent distance-$d$-dominating sets.
Abstract
We say a proper coloring of a graph is distance- fall if every vertex is within distance of at least one vertex of every color. We show that if is a connected graph of order at least that is -colorable, thenit has a distance-2 fall 3-coloring. Further, for every integer , if is a tree of order at least , then has a -coloring such that every vertex is within distance of every color. This proves an old conjecture of Beineke and Henning that every tree of order has an independent distance--dominating set of size at most .
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Taxonomy
Topicsgraph theory and CDMA systems · Graph Labeling and Dimension Problems · Advanced Graph Theory Research
A Note on Distance-Fall Colorings
Wayne Goddard
Clemson University, USA
Sonwabile Mafunda
Soka University of America, USA
University of Johannesburg, South Africa
Abstract
We say a proper coloring of a graph is distance- fall if every vertex is within distance of at least one vertex of every color. We show that if is a connected graph of order at least that is -colorable, then it has a distance-2 fall 3-coloring. Further, for every integer , if is a tree of order at least , then has a -coloring such that every vertex is within distance of every color. This proves an old conjecture of Beineke and Henning that every tree of order has an independent distance--dominating set of size at most .
Keywords: Fall coloring; distance- domination; chromatic number
MSC-class: 05C12; 05C15; 05C69
1 Introduction
A fall coloring is a proper coloring of the vertices of a graph such that every vertex sees every color. That is, for each vertex its closed neighborhood contains all colors. Not all graphs have a fall coloring. The simplest example is the -cycle. In this paper we consider proper colorings where every vertex is “near” to every color. We say a coloring is distance- fall if every vertex is within distance of at least one vertex of every color. For example, for a graph of diameter , every proper coloring is distance- fall; and every odd cycle has a -coloring that is distance- fall.
The terminology “fall coloring” was introduced in the 2000 paper by Dunbar et al. [5]. But the concept is older, having been studied before as partitioning a graph into independent dominating sets; see the references of [5]. Recall that a set of vertices is independent if no two elements of are joined by an edge. A set is dominating if every vertex is either in or adjacent to at least one vertex of . An independent dominating set is one that is both independent and dominating. More generally, a set is distance- dominating if every vertex is within distance of at least one vertex of . Thus a distance- fall coloring is a partition of the vertex set into independent distance- dominating sets. The parameter independent distance- domination, that is, the minimum size of an independent distance- dominating set, was originally studied by Beineke and Henning [1].
2 Distance-Fall Colorings and Chromatic Number
It is immediate that if a (connected nontrivial) graph is bipartite, then the bipartite coloring is a fall coloring. So the next question to consider is -colorable graphs. For a coloring, we say a vertex is -good if every color appears within distance of the vertex; otherwise the vertex is -bad. The following theorem shows that if the graph is -colorable then there is a proper coloring with colors such that every vertex is -good.
Theorem 2.1**.**
If is a connected graph of order at least that is -colorable, then has a distance- fall -coloring.
Proof: Suppose to the contrary, that does not admit a distance- fall -coloring. Take a proper -coloring of that minimizes the number of -bad vertices. Without loss of generality, choose a -bad vertex . Then is monochromatic.
If there exists a vertex with , then by recoloring with the missing color in , we obtain a proper -coloring of with strictly fewer -bad vertices, contradicting the minimality of the original coloring.
Therefore, assume for all . Since is -bad, we have and is monochromatic for every . That is, and all vertices in its second neighborhood share the same color.
Now recolor with the missing color in . Vertex then becomes -good, and every neighbor of remains -good. Moreover, no previously -good vertex becomes -bad under this recoloring. Thus, the total number of -bad vertices decreases, contradicting the minimality of the original coloring.
Therefore, there exists a proper -coloring of with zero -bad vertices.
It is perhaps interesting to note that an independent distance- dominating set is related to an independent isolating set. An isolating (or vertex-edge dominating) set can be viewed as a -dominating set with the added condition that if vertex is at distance from , then all its neighbors are at distance from . An independent isolating set is an isolating set that is also independent. See for example [2, 3, 6]. It is easy to see that Theorem 2.1 does not extend to a partition into three independent isolating sets (for example ). However, in [3] it was shown that it almost does: specifically, if is -colorable there exist three sets , , and of vertices, such that each is an independent isolating set, their union is all vertices, and such that and overlap in at most one vertex, while is disjoint from .
It is, however, unclear if the Theorem 2.1 generalizes to larger number of colors. The next step would be to determine whether every connected -colorable graph of order at least has a -coloring such that every vertex is within distance three of all colors. This is best possible because of path–complete graph, meaning the graph that consists of a path and a complete graph joined by an edge. However, in the case of trees, the above theorem does generalize:
Theorem 2.2**.**
For integer , if is a tree of order at least , then has a -coloring such that every vertex is within distance of every color.
Proof: Let be a tree of diameter . If , then any coloring that uses each color at least once automatically has the desired property.
So, assume . Let be a diametral path in . Define a coloring on the vertices of such that .
Now, let be a central edge of , and let and be the components of containing and , respectively. If denotes the forest induced by the vertices of not contained in , then color every vertex of such that, in each of the components and , any two vertices at the same distance from or receive the same color.
We claim that this coloring has the desired property. If a vertex lies within distance of the edge , then is within distance of each color on . Otherwise, the unique path from to enters at some vertex and continues along via ; the first vertices on this path all have distinct colors. Hence the theorem follows.
Theorem 2.1 is equivalent to saying that the vertices of a -colorable graph can be partitioned into three (disjoint) independent distance- dominating sets. And thus the independent distance- domination number of a -colorable graph is at most , where is the order. This generalizes Theorem 2 of [1], which proved the bound for trees. Furthermore, Theorem 2.2 shows that the independent distance- domination number of a tree is at most , which establishes the Conjecture at the end of [1].
We note that Theorem 2.2 and thus the conjecture in [1] was recently also proved by Bujtás et al. [4]. Indeed, they showed that Theorem 2.2 generalizes to bipartite graphs.
There is a version of Theorem 2.1 for general graphs with a slightly weaker distance condition. A partial coloring means a proper coloring where only some of the vertices are colored.
Theorem 2.3**.**
If is a connected graph of order at least , then has a partial -coloring such that every vertex is within distance of every color.
Proof: Consider any partial -coloring that maximizes the number of colored vertices. Note that any uncolored vertex has neighbors of every color, since it is not possible to color . Thus, out of all partial -colorings with the maximum number of colored vertices, take the one with the minimum number of -bad vertices and let be such a vertex.
We know is colored. Furthermore, all of its neighbors are colored. If some vertex at distance from is uncolored, then is -good since contains all colors. So assume every vertex within distance two of is colored. Then, as in the proof of Theorem 2.1, one can re-color either or a neighbor of so that and its neighbors are -good. Note that if this recoloring makes an uncolored vertex bad, then we have a contradiction of the original requirement that the maximum number of vertices were colored. So every vertex that is -bad is -good; or in other words, every vertex is -good.
This strengthens the first result in [1] for the case that .
We conclude with a brief comment about graph operations. Kaul and Mitillos [7] showed that if a graph has a fall -coloring and a graph has a -coloring, then the cartesian product has a fall -coloring. The same idea works here: if has a distance- fall -coloring and has a -coloring , then the cartesian product has a distance- fall -coloring. As per usual in the cartesian product, consider both colorings as assigning integers in the range to and take the coloring of vertex in the product to be the sum modulo .
We note further that if has a distance- fall -coloring and has a distance- fall -coloring, then the cartesian product has a distance- fall -coloring. Simply take the coloring of vertex in the product to be the ordered pair .
Similar results can be shown for other products.
Acknowledgment
The authors would like to thank Ben Gobler for suggesting the concept.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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