A Periodic packing coloring of some distance graphs

On Packing Colorings of Distance Graphs

Abstract

The packing chromatic number of a graph is the least integer for which there exists a mapping from to such that any two vertices of color are at a distance of at least . This paper studies the packing chromatic number of infinite distance graphs , i.e. graphs with the set of integers as vertex set, with two distinct vertices being adjacent if and only if . We present lower and upper bounds for , showing that for finite , the packing chromatic number is finite. Our main result concerns distance graphs with for which we prove some upper bounds on their packing chromatic numbers, the smaller ones being for : if is odd and if is even.

Abstract

Keywords:

graph coloring; packing chromatic number; distance graph.

1 Introduction

Let be a connected graph and let be an integer, . A packing -coloring (or simply a packing coloring) of a graph is a mapping from to such that for any two distinct vertices and , if , then , where is the distance between and in (thus vertices of color form an -packing of ). The packing chromatic number of is the smallest integer for which has a packing -coloring. This parameter was introduced recently by Goddard et al. [9] under the name of broadcast chromatic number and the authors showed that deciding whether is NP-hard. Fiala and Golovach [6] showed that the packing coloring problem is NP-complete for trees. Brešar et al. [2] studied the problem on Cartesian products graphs, hexagonal lattice and trees, using the name of packing chromatic number. Other studies on this parameter mainly concern infinite graphs, with a natural question to be answered : does a given infinite graph have finite packing chromatic number ? Goddard et al. answered this question affirmatively for the infinite two dimensional square grid by showing . The lower bound was later improved to by Fiala et al. [7] and then to by Ekstein et al. [5]. The upper bound was recently improved by Holub and Soukal [13] to . Fiala et al. [7] showed that the infinite hexagonal grid has packing chromatic number 7; while both the infinite triangular lattice and the 3-dimensional square lattice were shown to admit no finite packing coloring by Finbow and Rall [8]. Infinite product graphs were considered by Fiala et al. [7] who showed that the product of a finite path (of order at least two) with the 2-dimensional square grid has infinite packing chromatic number while the product of the infinite path and any finite graph has finite packing chromatic number. The (infinite) distance graph with distance set , where are positive integers, has the set of integers as vertex set, with two distinct vertices being adjacent if and only if . The finite distance graph is the subgraph of induced by vertices . To simplify, will also be denoted as and as . The study of distance graphs was initiated by Eggleton et al. [3]. A large amount of work has focused on colorings of distance graphs [4, 15, 1, 11, 12, 14], but other parameters have also been studied on distance graphs, like the feedback vertex set problem [10]. The aim of this paper is to study the packing chromatic number of infinite distance graphs, with particular emphasis on the case . In Section 2, we bound the packing chromatic number of the infinite path power (i.e. infinite distance graph with ). Section 3 concerns packing colorings of distance graphs with , for which we prove some lower and upper bounds on the number of colors (see Proposition 1). Exact or sharp results for the packing chromatic number of some other 4-regular distance graphs are presented in Section 4. Section 5 concludes the paper with some remarks and open questions. Our results about the packing chromatic number of for some small values of (from Sections 2 and 4) are summarized in Table 1.

Table 1: Lower and upper bounds for the packing chromatic number of for different values of . In the fourth column are the periods of the colorings giving the upper bounds. (: bound obtained by running Algorithm 4 of Section 4).

The bounds of Section 3 are summarized in the following Proposition:

Proposition 1.

Let be integers. Then,

Some proofs of lower bounds use a density argument. For this, we define the density of a color in as the maximum fraction of vertices colored in any packing coloring of and (or simply , if the graph is clear from the context) by . Let also be the maximum fraction of vertices colored or in any packing coloring of and let . We have trivially, for any , and .

2 Path Powers

The power of a graph is the graph with the same vertex set as and edges between every vertices that are at a mutual distance of at most in . Let be the power of the two-ways infinite path and let be the power of the path on vertices. We first present an asymptotic result on the packing chromatic number:

Proposition 2.

and .

Proof.

is a spanning subgraph of the lexicographic product1 (see Figure 1). Then, as Goddard et al. [9] showed that , the same upper bound holds for . To prove the lower bound, since , then for any packing coloring of using at most colors, must satisfy:

Since

where is the harmonic number and since , then implies . ∎

Figure 1: The infinite distance graph as a subgraph of the lexicographic product .
Corollary 1.

For any finite subset of , the packing chromatic number of is finite.

For very small , exact values or sharp bounds for the packing chromatic number can be calculated:

Proposition 3.
Proof.

A packing 8-coloring can be constructed by repeating the following pattern of length :

On the other hand, it can be seen that for any . However, we next prove that . Consider vertices for some . The only possibility to color more than 5 of these vertices is to give color 1 to and then at most 2 vertices can be given color 2 ( or , and or ). But in this case, neither vertex nor vertex can be given color 1 or 2, resulting in vertices colored out of . Moreover, an easy computation gives that

Proposition 4.
Proof.

The upper bound comes from a packing -coloring of period defined by repeating the sequence of length given in Appendix A. To prove the lower bound, as the distance between the vertices and is , then and an easy computation gives that

3 with large

The general method is to cut the distance graph into sets of consecutive vertices of size or , depending on the value of and to color each set by a predefined color pattern.

Figure 2: , with (on the top) and (on the bottom) drawn by rows of size .

Let be either or and let and be the subgraph of induced by . Notice that and that if , then each is an induced cycle of of length (see Figure 2). By a color pattern , we mean a sequence of integers of length that will be associated to some subgraph by giving the color to the vertex of . If is a sequence of integers, is the sequence obtained by repeating times. The cyclic distance between elements and of a sequence is . We first need to know the distance between two vertices in .

Lemma 1.

The distance between two vertices and of is , where , with .

Proof.

Let us call an edge joining vertices and , with a -edge. Assume, without loss of generality, that . then, any minimal path between and uses either -edges and -edges or -edges and -edges. ∎

The key lemma of our method is the following one which gives conditions for a coloring of by color patterns to be a packing coloring.

Lemma 2.

Let be a positive integer and for each integer , set . Let be a positive integer and for each , let be the subgraph of induced by , and be the graph with an additional edge joining vertices and if . Suppose that is colored in such a way that:

i)

for each integer , the coloring inherited by each is a packing coloring;

ii)

for each pair of integers and , if is the maximum common color in both and then we have , , and each that is a common color in both and has the property that is colored if and only if is colored for each .

Then the coloring is a packing coloring of whenever is in .

Proof.

Suppose vertices and have the same color, say , and, without loss of generality, assume is in . Let be defined by for each . Observe that when or , if two vertices and are adjacent in , then and are adjacent in . But then a path in between and maps via to a path of at most the same length between two vertices in colored . Since, by hypothesis, is colored by a packing coloring, as long as , the distance between and must be greater than . If , then for some . If , then and by Lemma 1, since by hypothesis, and . Else, if then and by Lemma 1, by hypothesis. ∎

3.1 Proof of Proposition 1

Proof.

Let be an integer, and if or for some ; if or . For each integer , set and let be the subgraph of induced by . In each of the following cases, a packing coloring of is defined by assigning to each subgraph a pattern of colors with length . We will use the following sub-patterns of colors: ,
,
,
,
,
,

            .

Case A. is odd.

First, since for some integer and thanks to Lemma 2, we can assign to each subgraph the color pattern . In order to color subgraphs , we consider three sub-cases (that are not totally disjoints).

Subcase A.1. for some .

A packing coloring of using these sub-patterns is constructed by assigning inductively to consecutive subgraphs the sequence of color patterns Since the cyclic distance between two occurrences of any color in each color pattern is always greater than , then Condition i) of Lemma 2 is satisfied. Moreover, as the cyclic distance between any two color patterns in is always greater than a quarter (since color patterns of are associated only with subgraphs of even indices) of their maximum common color, then Condition ii) is also satisfied. Hence, the coloring is a packing coloring of and .

Subcase A.2. for some .

We denote by any sequence obtained by inserting quasi evenly cyclically-distributed occurrences of the pair in the sequence ; insertions being made only after a color different from , in order to keep the sequence alternate between color 1 and other colors.
For example, can be rewritten as
. Then, color patterns using colors from are defined by: , for , ;
, for , ;
, for , ;
, for ;
, for ; and we assign inductively to consecutive subgraphs the sequence of color patterns defined by
In order for a color pattern to satisfy Condition of Lemma 2 and as the pairs have to be inserted only on even positions, we must have . Hence the worst case for this separation constraint is for color in when : one can insert occurrences of if , which is true as soon as and thus . Moreover, it can be seen that the added color in each pattern is chosen in such a way that Condition is satisfied. Hence, the coloring is a packing coloring of and .

Subcase A.3. for some , .

The base case is for which the sequence of color patterns that is assigned inductively to consecutive subgraphs is defined as follows: with , , , , and . As for Subcase A.1, it can be easily checked that the defined coloring is a packing coloring. Now, for , we may replace each of the above color patterns by a certain number of patterns (depending on the residue of modulo the length of the sub-pattern used) that will be used in turn, as for Subcase A.2. Let be the empty sequence and let and , be some integers (that will be set just after). Set , with , , , and

Set , with , , , and
Set , with , , , and
Set , with , , , and
Set , with , , , and
As the cyclic distance between two occurrences of either the color pattern or of or of in is equal to 4 (hence, each of these three patterns appears every 8 set ), and if is the maximum color used in , then, according to Lemma 2, for , must satisfy
Similarly, the cyclic distance between two occurrences of either the color pattern or of in is equal to 8, hence, for or , must satisfy Therefore, for each residue of modulo , a packing coloring is obtained by fixing the values of and as indicated in the next table ( is set to the smallest value satisfying the above inequations). The largest color used in each case is reported on the last row. 0 4 8 12 16 20 24 28 32 36 40 44 , / , 3 , 3 / , 3 , 3 / , 3 , 3 / , 3 , 3 , / , 3 , 3 / , 3 , 3 / , 3 , 3 / , 3 , 3 , / , 4 , 4 , 3 / , 4 , 3 , 4 / , 3 , 4 , 4 , / , 2 , 2 , 2 , 2 , 2 / , 2 , 2 , 2 , 2 , 2 , / , 2 , 2 , 2 / , 3 , 2 , 3 / , 2 , 2 , 2 largest color 31 59 59 59 51 69 41 75 47 49 63 89 An illustration for the case is given in Appendix B.

Case B. is even.

For or , recall that subgraphs are of size . New color patterns are constructed by inserting a new color at the end of each pattern (of length ) defined in Subcases A.1, A.2 and A.3. By Lemma 2, the problem of adding the missing color in each color pattern defined in subcases A.1, A.2 and A.3 is equivalent to the one of coloring the infinite path with colors from such that vertices of color are at distance greater than . We are going to show, by induction on , that . For , vertices can be colored by alternating color and color , so . Assume that can be colored with colors from and let . Replace now color by colors and alternatively. Then the largest color used is and the constraint is satisfied since if vertices and are colored then their mutual distance satisfies . As the colorings defined in Subcase A.1 (Subcases A.2 and A.3, respectively) use colors from to ( and at most , respectively), then we obtain a packing coloring of with colors from to at most ( and , respectively), provided that ( and , respectively). ∎

Remark 1.
  • In Subcase A.2, the method can produce a packing coloring using less than colors, depending on the value of (i.e. if some are equal to zero).

  • A combination of the methods of Subcases A.2 and A.3 could be used to define a packing coloring for odd , , using less colors than in Subcase A.3.

  • For Case B, it seems that less than colors are sufficient for such a coloring. When , a computation gives for such a coloring; when , we find and when , we find .

4 with small and

The results from Section 3 do not apply for with small , however it is possible to derive exact or sharp results for some of them, using density arguments and the computer. Algorithm 4 is a simple algorithm that prints all the packing -colorings of . It checks, for each vertex, each possible color in a recursive fashion. Hence it must be used by initializing the first elements of the array color to and calling . {algorithm}[ht] \KwDataglobal integers ; global array color; \If print(color)  \Else \For from 1 \KwTo \If such that color[]= and color[]   RecColor()  color[] 0  RecColor()

Proposition 5.
Proof.

first, remark that the graph-distance between vertex and vertex is . A packing 9-coloring of of period is given by the following sequence: