OOn Packing Colorings of Distance Graphs
Olivier Togni
LE2I, UMR CNRS 5158Universit´e de Bourgogne, 21078 Dijon cedex, France
November 2, 2018
The packing chromatic number χ ρ ( G ) of a graph G is the least integer k forwhich there exists a mapping f from V ( G ) to { , , . . . , k } such that any twovertices of color i are at a distance of at least i + 1. This paper studies the packingchromatic number of infinite distance graphs G ( Z , D ), i.e. graphs with the set Z of integers as vertex set, with two distinct vertices i, j ∈ Z being adjacent ifand only if | i − j | ∈ D . We present lower and upper bounds for χ ρ ( G ( Z , D )),showing that for finite D , the packing chromatic number is finite. Our mainresult concerns distance graphs with D = { , t } for which we prove some upperbounds on their packing chromatic numbers, the smaller ones being for t ≥ χ ρ ( G ( Z , { , t } )) ≤
40 if t is odd and χ ρ ( G ( Z , { , t } )) ≤
81 if t is even. Keywords: graph coloring; packing chromatic number; distance graph.
1. Introduction
Let G be a connected graph and let k be an integer, k ≥
1. A packing k -coloring (or simplya packing coloring) of a graph G is a mapping f from V ( G ) to { , , · · · , k } such that for anytwo distinct vertices u and v , if f ( u ) = f ( v ) = i , then dist( u, v ) > i , where dist( u, v ) is thedistance between u and v in G (thus vertices of color i form an i -packing of G ). The packingchromatic number χ ρ ( G ) of G is the smallest integer k for which G has a packing k -coloring.This parameter was introduced recently by Goddard et al. [9] under the name of broadcastchromatic number and the authors showed that deciding whether χ ρ ( G ) ≤ does a given infinite graphhave finite packing chromatic number ? Goddard et al. answered this question affirmativelyfor the infinite two dimensional square grid by showing 9 ≤ χ ρ ≤
23. The lower bound waslater improved to 10 by Fiala et al. [7] and then to 12 by Ekstein et al. [5]. The upper bound1 a r X i v : . [ c s . D M ] F e b as recently improved by Holub and Soukal [13] to 17. Fiala et al. [7] showed that the infinitehexagonal grid has packing chromatic number 7; while both the infinite triangular lattice andthe 3-dimensional square lattice were shown to admit no finite packing coloring by Finbowand Rall [8]. Infinite product graphs were considered by Fiala et al. [7] who showed that theproduct of a finite path (of order at least two) with the 2-dimensional square grid has infinitepacking chromatic number while the product of the infinite path and any finite graph hasfinite packing chromatic number.The (infinite) distance graph G ( Z , D ) with distance set D = { d , d , . . . , d k } , where d i arepositive integers, has the set Z of integers as vertex set, with two distinct vertices i, j ∈ Z being adjacent if and only if | i − j | ∈ D . The finite distance graph G n ( D ) is the subgraphof G ( Z , D ) induced by vertices 0 , , . . . , n −
1. To simplify, G ( Z , { d , d , . . . , d k } ) will also bedenoted as D ( d , d , . . . , d k ) and G n ( { d , d , . . . , d k } ) as D n ( d , d , . . . , d k ).The study of distance graphs was initiated by Eggleton et al. [3]. A large amount of workhas focused on colorings of distance graphs [4, 15, 1, 11, 12, 14], but other parameters havealso 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 D = { , t } . In Section 2, we bound the packing chro-matic number of the infinite path power (i.e. infinite distance graph with D = { , , . . . , t } ).Section 3 concerns packing colorings of distance graphs with D = { , t } , for which we provesome lower and upper bounds on the number of colors (see Proposition 1). Exact or sharp re-sults for the packing chromatic number of some other 4-regular distance graphs are presentedin Section 4. Section 5 concludes the paper with some remarks and open questions.Our results about the packing chromatic number of G ( Z , D ) for some small values of D (from Sections 2 and 4) are summarized in Table 1. D χ ρ ≥ χ ρ ≤ period1 , , ∗ , , ∗
12 10281 , , ∗
15 6401 , ∗
25 51841 , ∗
18 5761 , , , , G ( Z , D ) for differentvalues of D . In the fourth column are the periods of the colorings giving the upperbounds. ( ∗ : bound obtained by running Algorithm 1 of Section 4).The bounds of Section 3 are summarized in the following Proposition:2 roposition 1. Let t, q be integers. Then, χ ρ ( D (1 , t )) ≤ , t = 2 q + 1 , q ≥ , t = 2 q + 1 , q ≥ , t = 2 q, q ≥ , t = 2 q, q ≥ , t = 96 q ± , q ≥ , t = 96 q + 1 ± , q ≥ . Some proofs of lower bounds use a density argument. For this, we define the density ρ a ( G n ( D )) of a color a in G n ( D ) as the maximum fraction of vertices colored a in anypacking coloring of G n ( D ) and ρ a ( D ) (or simply ρ a , if the graph is clear from the context)by ρ a ( D ) = lim sup n → + ∞ ρ a ( G n ( D )). Let also ρ , ( G n ( D )) be the maximum fraction of verticescolored 1 or 2 in any packing coloring of G n ( D ) and let ρ , = lim sup n → + ∞ ρ , ( G n ( D )). We havetrivially, for any D , χ ρ ( G ( Z , D )) ≥ min { c | (cid:80) ci =1 ρ i ≥ } and ρ , ≤ ρ + ρ .
2. Path Powers
The t th power G t of a graph G is the graph with the same vertex set as G and edges betweenevery vertices x, y that are at a mutual distance of at most t in G . Let D t = G ( Z , { , , · · · , t } )be the t th power of the two-ways infinite path and let P tn = G n ( { , , · · · , t } ) be the t th powerof the path P n on n vertices.We first present an asymptotic result on the packing chromatic number: Proposition 2. χ ρ ( D t ) = (1 + o (1))3 t and χ ρ ( D t ) = Ω( e t ) .Proof. D t is a spanning subgraph of the lexicographic product Z ◦ K t (see Figure 1). Then,as Goddard et al. [9] showed that χ ρ ( Z ◦ K t ) = (1 + o (1))3 t , the same upper bound holds for D t . To prove the lower bound, since ρ i ≤ it +1 , then for any packing coloring of D t using atmost c colors, c must satisfy: c (cid:88) i =1 it + 1 ≥ . Since c (cid:88) i =1 it + 1 < c (cid:88) i =1 it = 1 t c (cid:88) i =1 i = H c t , where H n is the n th harmonic number and since H n = Ω(ln( n )), then H c t ≥ c = Ω( e t ). Corollary 1.
For any finite subset D of N , the packing chromatic number of G ( Z , D ) isfinite. For very small t , exact values or sharp bounds for the packing chromatic number can becalculated: the lexicographic product G ◦ H of graphs G and H has vertex set V ( G ) × V ( H ) and two vertices ( a, x ) and( b, y ) are linked by an edge if and only if ab ∈ E ( G ) or a = b and xy ∈ E ( H ) D as a subgraph of the lexicographic product Z ◦ K . Proposition 3. χ ρ ( D ) = 8 . Proof.
A packing 8-coloring can be constructed by repeating the following pattern of length54 : 8 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . On the other hand, it can be seen that ρ i ≤ i +1 for any i ≥
1. However, we next provethat ρ , ≤ . Consider vertices v, v + 1 , . . . , v + 9 for some v . The only possibility to colormore than 5 of these 10 vertices is to give color 1 to v, v + 3 , v + 6 , v + 9 and then at most 2vertices can be given color 2 ( v + 1 or v + 2, and v + 7 or v + 8). But in this case, neithervertex v + 10 nor vertex v + 11 can be given color 1 or 2, resulting in 6 vertices colored out of12. Moreover, an easy computation gives that χ ρ ( D ) ≥ min { c | + (cid:80) ci =3 12 i +1 ≥ } = 8 . Proposition 4. ≤ χ ρ ( D ) ≤ . Proof.
The upper bound comes from a packing 23-coloring of period 768 defined by repeatingthe sequence of length 768 given in Appendix A.To prove the lower bound, as the distance dist( u, v ) between the vertices u and v isdist( u, v ) = (cid:100) v − u (cid:101) , then ρ i ≤ i +1 and an easy computation gives that χ ρ ( D ) ≥ min { c | (cid:80) ci =1 13 i +1 ≥ } = 17 . D (1 , t ) with large t The general method is to cut the distance graph into sets of consecutive vertices of size s = t − s = t + 1, depending on the value of t and to color each set by a predefined colorpattern. Let s be either t + 1 or t − A i = { is, is + 1 , . . . , ( i + 1) s − } and B i be thesubgraph of D (1 , t ) induced by A i . Notice that V ( D (1 , t )) = (cid:83) + ∞ i = −∞ A i and that if s = t + 1,then each B i is an induced cycle of D (1 , t ) of length s = t + 1 (see Figure 2). By a colorpattern P , we mean a sequence of integers ( c , c , . . . , c s ) of length s that will be associated tosome subgraph B i by giving the color c j to the j th vertex of B i . If S is a sequence of integers, S p is the sequence obtained by repeating S p times. The cyclic distance between elements s i and s j of a sequence ( s , s , . . . , s (cid:96) ) is min( | j − i | , (cid:96) − | j − i | ).We first need to know the distance between two vertices in D (1 , t ).4 B B B B B B B s = | B i | = t + 1 s = | B i | = t − D (1 , D (1 , Figure 2: D (1 , t ), with t = 7 (on the top) and t = 9 (on the bottom) drawn by rows of size s = 8. Lemma 1.
The distance between two vertices u and v of D (1 , t ) is dist ( u, v ) = min( q + r, q +1 + t − r ) , where | v − u | = qt + r , with ≤ r < t .Proof. Let us call an edge joining vertices x and y , with | y − x | = k a k -edge. Assume, withoutloss of generality, that v ≥ u . then, any minimal path between u and v uses either q t -edgesand r q + 1 t -edges and t − r D (1 , t ) by color patterns to be a packing coloring. Lemma 2.
Let s > be a positive integer and for each integer i , set A i = { is, is + 1 , . . . , ( i +1) s − } . Let t be a positive integer and for each i , let B i be the subgraph of G = D (1 , t ) inducedby A i , and C i be the graph B i with an additional edge joining vertices is and ( i + 1) s − if s = t − . Suppose that G is colored in such a way that: i) for each integer i , the coloring inherited by each C i is a packing coloring; ii) for each pair of integers i and j , if c is the maximum common color in both C i and C j then we have c < s , | i − j | > c , and each b ≤ c that is a common color in both C i and C j has the property that si + k is colored b if and only if sj + k is colored b for each k ∈ { , , . . . , s − } .Then the coloring is a packing coloring of G whenever t is in { s + 1 , s − } . roof. Suppose vertices u and v have the same color, say e , and, without loss of generality,assume u is in B . Let σ : V ( G ) → V ( C ) be defined by σ ( k ) = k mod s for each k ∈ N .Observe that when t = s + 1 or t = s −
1, if two vertices x and y are adjacent in G , then σ ( x ) and σ ( y ) are adjacent in C . But then a path in G between u and v maps via σ to apath of at most the same length between two vertices in C colored e . Since, by hypothesis, C is colored by a packing coloring, as long as u (cid:54) = σ ( v ), the distance between u and v mustbe greater than e .If u = σ ( u ) = σ ( v ), then v − u = js for some j . If s = t −
1, then v − u = j ( t −
1) =( j − t + t − j and by Lemma 1, dist( u, v ) = min( j − t − j, j + t − t + j ) = min( t − , j ) > e since by hypothesis, e < s = t − j > e . Else, if s = t + 1 then v − u = j ( t + 1) = jt + j and by Lemma 1, dist( u, v ) = min( j + j, j + 1 + t − j ) = min(2 j, t + 1) > e by hypothesis. Proof.
Let t be an integer, G = D (1 , t ) and s = 4 p if t = 4 p − t = 4 p + 1 for some p ; s = 4 p + 1 if t = 4 p or t = 4 p + 2. For each integer i , set A i = { is, is + 1 , . . . , ( i + 1) s − } and let B i be the subgraph of G induced by A i .In each of the following cases, a packing coloring of G is defined by assigning to eachsubgraph B i a pattern of colors with length s . We will use the following sub-patterns ofcolors: S , = (1 , , , S , = (1 , , , , , , , , , , , S , = (1 , , , , , , , , , , , S , = (1 , , , , , , , , , , , , , , , S , = (1 , , , , , , , , , , , , , , , , , , , , , , , S , = (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , S , = (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Case A. t is odd. First, since s = 4 p for some integer p and thanks to Lemma 2, we canassign to each subgraph B i +1 the color pattern ( S , ) p . In order to color subgraphs B i , weconsider three sub-cases (that are not totally disjoints). Subcase A.1. t = 96 q ± for some q ≥ . A packing coloring of D (1 , t ) using these sub-patterns is constructed by assigning inductively to 8 consecutive subgraphs B i the sequenceof color patterns P = (( S , ) q , ( S , ) q , ( S , ) q , ( S , ) q , ( S , ) q , ( S , ) q , ( S , ) q , ( S , ) q ) . Since the cyclic distance between two occurrences of any color e in each color pattern isalways greater than e , then Condition i) of Lemma 2 is satisfied. Moreover, as the cyclicdistance between any two color patterns in P is always greater than a quarter (since colorpatterns of P are associated only with subgraphs of even indices) of their maximum commoncolor, then Condition ii) is also satisfied. Hence, the coloring is a packing coloring of D (1 , t )and χ ρ ( D (1 , t )) ≤
29. 6 ubcase A.2. t = 2 p + 1 for some p ≥ . We denote by S (1 , α ) r any sequence obtainedby inserting r quasi evenly cyclically-distributed occurrences of the pair (1 , α ) in the sequence S ; insertions being made only after a color different from 1, in order to keep the sequencealternate between color 1 and other colors.For example, (1 , , , , , , , , , , , (1 , α ) can be rewritten as (1 , , , , , , , α, , , , , , , , α, , , , , , , , , , α, , , , , , , , , , α, , , , , , , , , , α ). Then, color patterns using colors from { , , . . . , } are defined by: Q i = ( S , ) q (1 ,
32 + i ) r , for s = 12 q + 2 r , ≤ r ≤ i = 0 , , Q i = ( S , ) q (1 ,
35 + i ) r , for s = 12 q + 2 r , ≤ r ≤ i = 0 , , Q i = ( S , ) q (1 ,
38 + i ) r , for s = 16 q + 2 r , ≤ r ≤ i = 0 , , Q = ( S , ) q (1 , r , for s = 24 q + 2 r , ≤ r ≤ Q = ( S , ) q (1 , r , for s = 32 q + 2 r , ≤ r ≤ B i the sequence of color patterns Q defined by Q = ( Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q , Q ) . In order for a color pattern S (1 , α ) r to satisfy Condition i ) of Lemma 2 and as thepairs (1 , α ) have to be inserted only on even positions, we must have 2 (cid:98)(cid:98) | S | r (cid:99) / (cid:99) ≥ α . Hencethe worst case for this separation constraint is for color 31 in Q when r = 14: one caninsert 14 occurrences of (1 ,
31) if 2 (cid:98)(cid:98) q (cid:99) / (cid:99) ≥
31, which is true as soon as q = 14 and thus s = 448. Moreover, it can be seen that the added color in each pattern is chosen in such away that Condition ii ) is satisfied. Hence, the coloring is a packing coloring of D (1 , t ) and χ ρ ( D (1 , t )) ≤ Subcase A.3. t = 2 p + 1 for some p , ≤ p ≤ . The base case is s ≡ B i is defined as follows: R = ( R , R , R , R , R , R , R , R ) , with R = ( S , ) q , R = ( S , ) q , R = ( S , ) q , R = ( S , ) q , and R = ( S , ) q .As for Subcase A.1, it can be easily checked that the defined coloring is a packing coloring.Now, for s (cid:54)≡ R j ∈ R bya certain number of patterns R ij (depending on the residue of s modulo the length of thesub-pattern used) that will be used in turn, as for Subcase A.2.Let (cid:15) be the empty sequence and let c j and δ j , 1 ≤ j ≤ R i = ( S , ) q .T i , with s = 12 q + 4 r , 0 ≤ r <
3, 0 ≤ i < δ , and T i = (cid:15), if r = 0;(1 , c + i, , c + δ + i ) , if r = 1;(1 , , , , , c + i, , c + δ + i ) , if r = 2 . Set R i = ( S , ) q .T i , with s = 12 q + 4 r , 0 ≤ r <
3, 0 ≤ i < δ , and T i = (cid:15), if r = 0;(1 , c + i, , c + δ + i ) , if r = 1;(1 , , , , , c + i, , c + δ + i ) , if r = 2 . R i = ( S , ) q .T i , with s = 16 q + 4 r , 0 ≤ r <
4, 0 ≤ i < δ , and T i = (cid:15), if r = 0;(1 , c + i, , c + δ + i ) , if r = 1;(1 , , , , , c + i, , c + δ + i ) , if r = 2;(1 , , , , , c + i, , c + δ + i, , c + 2 δ + i, , c + 3 δ + i ) , if r = 3 . Set R i = ( S , ) q .T i , with s = 24 q + 4 r , 0 ≤ r <
6, 0 ≤ i < δ , and T i = (cid:15), if r = 0;(1 , c + i, , c + δ + i ) , if r = 1;(1 , , , , , c + i, , c + δ + i ) , if r = 2;(1 , , , , , c + i, , c + δ + i, , c + 2 δ + i, , c + 3 δ + i ) , if r = 3;(1 , , , , , c + i, , c + δ + i, , , , , , c + 2 δ + i, , c + 3 δ + i ) , if r = 4;(1 , , , , , c + i, , c + δ + i, , , , , , c + 2 δ + i, , c + 3 δ + i, , c + 4 δ + i, , c + 5 δ + i ) , if r = 5;Set R i = ( S , ) q − .T i , with s = 48 q + 4 r , 0 ≤ r <
12, 0 ≤ i < δ , and T i = S , , if r = 0; S , . (1 , c + i, , c + δ + i ) , if r = 1; S , . (1 , , , , , c + i, , c + δ + i ) , if r = 2; S , . (1 , , , , , c + i, , c + δ + i, , c + 2 δ + i, , c + 3 δ + i ) , if r = 3;( S , ) , if r = 4;( S , ) . (1 , c + i, , c + δ + i ) , if r = 5;( S , ) . (1 , , , , , c + i, , c + δ + i ) , if r = 6;( S , ) . (1 , , , , , c + i, , c + δ + i, , c + 2 δ + i, , c + 3 δ + i ) , if r = 7; S , .S , , if r = 8; S , .S , . (1 , c + i, , c + δ + i ) , if r = 9; S , .S , . (1 , , , , , c + i, , c + δ + i ) , if r = 10; S , .S , . (1 , , , , , c + i, , c + δ + i, , c + 2 δ + i, , c + 3 δ + i ) , if r = 11;As the cyclic distance between two occurrences of either the color pattern R or of R orof R in R is equal to 4 (hence, each of these three patterns appears every 8 set B i ), and if e is the maximum color used in R ij , then, according to Lemma 2, for j = 1 , , δ j must satisfy δ j ≥ , if e ≤ , if 16 ≤ e ≤ , if 32 ≤ e ≤ , if 48 ≤ e ≤ , if 64 ≤ e ≤ . Similarly, the cyclic distance between two occurrences of either the color pattern R or of R in R is equal to 8, hence, for j = 4 or 5, δ j must satisfy δ j ≥ , if e ≤ , if 32 ≤ e ≤ , if 64 ≤ e ≤ . Therefore, for each residue of s modulo 48, a packing coloring is obtained by fixing thevalues of c j and δ j as indicated in the next table ( δ j is set to the smallest value satisfying theabove inequations). The largest color used in each case is reported on the last row.8 (mod 48) 0 4 8 12 16 20 24 28 32 36 40 44 c , δ / 32, 3 32, 3 / 32, 3 32, 3 / 32, 3 32, 3 / 32, 3 32, 3 c , δ / 38, 3 38, 3 / 38, 3 38, 3 / 38, 3 38, 3 / 38, 3 38, 3 c , δ / 44, 4 44, 4 32, 3 / 44, 4 32, 3 44, 4 / 32, 3 44, 4 44, 4 c , δ / 52, 2 52, 2 44, 2 44, 2 52, 2 / 60, 2 44, 2 38, 2 52, 2 60, 2 c , δ / 56, 2 56, 2 52, 2 / 64, 3 38, 2 64, 3 / 46, 2 60, 2 78, 2largest color 31 59 59 59 51 69 41 75 47 49 63 89 An illustration for the case s ≡
28 (mod 48) is given in Appendix B.
Case B. t is even. For t = 4 p or t = 4 p + 2, recall that subgraphs B i are of size s = 4 p + 1.New color patterns are constructed by inserting a new color at the end of each pattern (oflength s (cid:48) = s − p ) 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 insubcases A.1, A.2 and A.3 is equivalent to the one of coloring the infinite path P ∞ with colorsfrom { k , k + 1 , . . . , k } such that vertices of color e are at distance greater than e .We are going to show, by induction on k , that k ≤ k −
1. For k = 2, vertices can becolored by alternating color 2 and color 3, so k = 3. Assume that P ∞ can be colored withcolors from { k , k + 1 , . . . , k ≤ k − } and let k (cid:48) = k + 1. Replace now color k by colors k + 1 and k + 2 alternatively. Then the largest color used is k (cid:48) = k + 2 ≤ k + 1 = 2 k (cid:48) − x and y are colored k + 2 then their mutualdistance satisfies dist( x, y ) > k ≥ k +12 > k .As the colorings defined in Subcase A.1 (Subcases A.2 and A.3, respectively) use colorsfrom 1 to 29 (40 and at most 89, respectively), then we obtain a packing coloring of D (1 , t )with colors from 1 to at most 2 × − t ≥ Remark 1. • In Subcase A.2, the method can produce a packing coloring using less than colors, depending on the value of s (i.e. if some r i are equal to zero). • A combination of the methods of Subcases A.2 and A.3 could be used to define a packingcoloring for odd t , ≤ t ≤ , using less colors than in Subcase A.3. • For Case B, it seems that less than k − colors are sufficient for such a coloring.When k = 90 , a computation gives k = 156 for such a coloring; when k = 41 , wefind k = 72 and when k = 30 , we find k = 53 . D ( a, b ) with small a and b The results from Section 3 do not apply for D (1 , t ) with small t , however it is possible toderive exact or sharp results for some of them, using density arguments and the computer.Algorithm 1 is a simple algorithm that prints all the packing k -colorings of D n (1 , t ). Itchecks, for each vertex, each possible color in a recursive fashion. Hence it must be used byinitializing the first n elements of the array color to 0 and calling RecColor(0). Proposition 5. χ ρ ( D (1 , . lgorithm 1: RecColor( i ) Data : global integers n, k, t ; global array color ; if i = n then print( color ); elsefor c from 1 to k doif (cid:64) j < i such that color [ j ]= c and dist ( i, j ) ≤ c then color [ i ] ← c ;RecColor( i + 1); color [ i ] ← Proof. first, remark that the graph-distance dist( i, j ) between vertex i and vertex j ≥ i isdist( i, j ) = (cid:98) j − i (cid:99) + ( j − i ) (mod 3).A packing 9-coloring of D (1 ,
3) of period 32 is given by the following sequence:1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . It is routine to check that the vertices of a same color are sufficiently distant. On the otherhand, running an implementation of Algorithm 1 with n = 100, k = 8, and t = 3, outputs nocoloring, showing that 8 colors are not sufficient for a packing coloring of D (1 , Proposition 6. ≤ χ ρ ( D (1 , ≤ ≤ χ ρ ( D (1 , ≤ ≤ χ ρ ( D (1 , ≤ ≤ χ ρ ( D (1 , ≤ ≤ χ ρ ( D (1 , ≤ ≤ χ ρ ( D (1 , ≤ . Proof.
For the upper bounds, packing k -colorings are defined by exhibiting a pattern usingcolors from { , · · · , k } of length (cid:96) for each case. For D (1 , k = 16 and (cid:96) =320 is given in Appendix A. For D (1 ,
5) ( D (1 , D (1 , D (1 , D (1 , k, (cid:96) ) = (12 , , , , , .For the lower bounds, we use either density arguments or computer running Algorithm 1.For D (1 , ρ ≤ since at most 2 out of 5 consecutive vertices can be colored 1.Moreover, ρ i ≤ i − for i ≥ { c | + (cid:80) ci =2 14 i − ≥ } = 11.For D (1 , n = 43, k = 9, and t = 5outputs no coloring. Hence χ ρ ( D (1 , ≥ D (1 , ρ ≤ since at most 3 out of 7 consecutive vertices can be colored1. We now show that ρ ≤ . Let v be a vertex colored 2. If v + 3 is also colored 2, thenno vertex of { v + 4 , · · · , v + 10 } can be colored 2. Hence 2 vertices out of 11 are colored 2.10f v + 3 is not colored 2 but v + 4 is, then only one of v + 8, v + 14 can be colored 2 among { v + 5 , · · · , v + 16 } , resulting in 3 out of 17 vertices colored 2 and < . If neither v + 3nor v + 4 is colored 2 then no vertex of { v + 5 , v + 6 , v + 7 } can be colored 2 and at most onevertex of { v + 8 , v + 9 , v + 10 } can have color 2, resulting in 2 vertices out of 11 colored 2.Moreover, if i ≥
3, then ρ i ≤ i − and min { c | + + (cid:80) ci =3 16 i − ≥ } = 12.For D (1 , n = 44, k = 9, and t = 7outputs no coloring. Hence χ ρ ( D (1 , ≥ D (1 , n = 41, k = 10, and t = 8outputs no coloring. Hence χ ρ ( D (1 , ≥ D (1 , n = 46, k = 9, and t = 9outputs no coloring. Hence χ ρ ( D (1 , ≥ D (1 ,
5) with the followingperiodic packing 13-coloring of period 80 (compared with the packing 12-coloring of period1028):1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . We now turn our attention to other 4-regular distance graphs, i.e. graphs of type D ( a, b ),with 2 ≤ a ≤ b . First, remark that if a and b are not co-prime, then the graph D ( a, b ) is notconnected and consists in g = gcd( a, b ) copies of D ( ag , bg ). Hence we only consider distancegraphs D ( a, b ) with gcd( a, b ) = 1.The smallest example is D (2 ,
3) which is a subgraph of D (1 , ,
3) = P ∞ , thus χ ρ ( D (2 , ≤ χ ρ ( P ∞ ) ≤
23. In fact, we show that the upper bound is much less than 23:
Proposition 7. ≤ χ ρ ( D (2 , ≤ ≤ χ ρ ( D (2 , ≤ . Proof.
The lower bound 11 ≤ χ ρ ( D (2 , ρ i of a color i : it can be seen that ρ = and ρ i = i +1 for i ≥ { c | + (cid:80) ci =2 13 i +1 ≥ } = 11.For the lower bound 14 ≤ χ ρ ( D (2 , ρ = and ρ i = i − for i ≥ { c | + (cid:80) ci =2 15 i − ≥ } = 14.The upper bounds come from the packing 13-coloring of D (2 ,
3) of period 240 and thepacking 23-coloring of D (2 ,
5) of period 336 given in Appendix A.
5. Concluding remarks
We have shown that the packing chromatic number of any infinite distance graph with finite D is finite and is at most 40 (81, respectively) for D = { , t } with t being an odd (even,respectively) integer greater than or equal to 447.Among the many possible research directions, one can try to find better bounds and/ormore simple methods for D (1 , t ). In fact, running a simple greedy packing coloring algorithmthat consists in coloring vertices of a distance graph one-by-one from the left to the right11
10 15 20 25 30 35 40 45 50 55 60 0 50 100 150 200 250 300 350 400 450 500 nu m b e r o f c o l o r s tgreedy packing algorithm for D(1,t) Figure 3: Number of colors for a packing coloring of D (1 , t ) using a greedy algorithm.with the smallest color with respect to the constraint, suggests that the upper bounds foundin Section 3 can be strengthened. Figure 3 shows the number of colors used by the greedyalgorithm for a packing coloring of D n (1 , t ) (with n = 1000000) as a function of t for thefirst 500 values of t . One can see on the figure that for large t , the algorithm finds a packingcoloring, using between 30 and 50 colors. Moreover, more colors are needed in general when t is even compared to when t is odd. But surprisingly, even if we look only at even (or odd)values of t , the function is not monotonic. We wonder if the same goes for χ ρ . An interestingfuture work would be to study in more details the behavior of this greedy algorithm.Finally, a summary of the values of t for which a upper bound on the the packing chromaticnumber of D (1 , t ) is known and those that remain to be found is presented in Table 2.odd t →
45 47 ,
49 51 →
69 71 →
445 447 → + ∞ χ ρ ≤ ? 31 ? between 29 and 89 40even t →
94 96 ,
98 100 →
142 144 →
446 448 → + ∞ χ ρ ≤ ? 59 ? between 59 and 179 81Table 2: Known upper bounds for the packing chromatic number of D (1 , t ) Acknowledgements
The author is very indebted to the anonymous referees for their careful reading of themanuscript and their accurate comments; in particular to one of them for suggesting thestatement and proof of Lemma 2 in its present form and many other changes. We wish alsoto thank Pˇremysl Holub for the valuable discussions and for his pertinent comments on apreliminary version of the paper. 12 eferences [1] J. Barajas and O. Serra. Distance graphs with maximum chromatic number.
DiscreteMath. , 308(8):1355–1365, 2008.[2] B. Breˇsar, S. Klavˇzar, and D. F. Rall. On the packing chromatic number of Cartesianproducts, hexagonal lattice, and trees.
Discrete Appl. Math. , 155(17):2303–2311, 2007.[3] R. B. Eggleton, P. Erd˝os, and D. K. Skilton. Colouring the real line.
J. Combin. TheorySer. B , 39(1):86–100, 1985.[4] R. B. Eggleton, P. Erd˝os, and D. K. Skilton. Colouring prime distance graphs.
GraphsCombin. , 6(1):17–32, 1990.[5] J. Ekstein, J. Fiala, P. Holub, and B. Lidick´y. The packing chromatic number of thesquare lattice is at least 12. arXiv:1003.2291v1, 2010.[6] J. Fiala and P. A. Golovach. Complexity of the packing coloring problem for trees.
Dis-crete Applied Mathematics , 158(7):771 – 778, 2010. Third Workshop on Graph Classes,Optimization, and Width Parameters Euge ne, Oregon, USA, October 2007.[7] J. Fiala, S. Klavˇzar, and B. Lidick´y. The packing chromatic number of infinite productgraphs.
European J. Combin. , 30(5):1101–1113, 2009.[8] A. S. Finbow and D. F. Rall. On the packing chromatic number of some lattices.
Dis-crete Applied Mathematics , 158(12):1224 – 1228, 2010. Traces from LAGOS’07 IV LatinAmerican Algorithms, Graphs, and Optimization Symposium Puerto Varas - 2007.[9] W. Goddard, S. M. Hedetniemi, S. T. Hedetniemi, J. M. Harris, and D. F. Rall. Broadcastchromatic numbers of graphs.
Ars Combin. , 86:33–49, 2008.[10] H. Kheddouci and O. Togni. Bounds for minimum feedback vertex sets in distance graphsand circulant graphs.
Discrete Math. Theor. Comput. Sci. , 10(1):57–70, 2008.[11] D. D.-F. Liu. From rainbow to the lonely runner: a survey on coloring parameters ofdistance graphs.
Taiwanese J. Math. , 12(4):851–871, 2008.[12] D. D.-F. Liu and X. Zhu. Fractional chromatic number of distance graphs generated bytwo-interval sets.
European J. Combin. , 29(7):1733–1743, 2008.[13] R. Soukal and P. Holub. A note on packing chromatic number of the square lattice.
Electronic Journal of Combinatorics , (N17), 2010.[14] J. Steinhardt. On coloring the odd-distance graph.
Electron. J. Combin. , 16(1):Note 12,7, 2009.[15] M. Voigt and H. Walther. Chromatic number of prime distance graphs.
Discrete Appl.Math. , 51(1-2):197–209, 1994. 2nd Twente Workshop on Graphs and CombinatorialOptimization (Enschede, 1991). 13 . Periodic packing coloring of some distance graphs
A periodic packing 23-coloring of D (1 , , of period , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , A periodic packing 16-coloring of D (1 , of period , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , A periodic packing 13-coloring of D (2 , of period , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , A periodic packing 23-coloring of D (2 , of period , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . An illustration of Subcase A.3 of the proof of Proposition 1 We illustrate the construction of a packing coloring of D (1 , t ) defined in Subcase A.3 for t = 75 or t = 77, i.e. s = 76 = 48 + 28.The color patterns R ij are defined as follows: R i = (1 , , , , , , , , , , , . (1 ,
32 + i, ,
35 + i ), 0 ≤ i ≤ R i = (1 , , , , , , , , , , , . (1 ,
38 + i, ,
41 + i ), 0 ≤ i ≤ R i = (1 , , , , , , , , , , , , , , , . (1 , , , , ,
44 + i, ,
48 + i, ,
52 + i, ,
56 + i ),0 ≤ i ≤ R i = (1 , , , , , , , , , , , , , , , , , , , , , , , . (1 ,
60 + i, ,
62 + i ) , ≤ i ≤ R i = (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . (1 , , , , ,
64 + i, ,
67 + i, ,
70 + i, ,
73 + i ), 0 ≤ i ≤ B i +1 the color pattern(1 , , , and repeatedly to 48 consecutive subgraphs B i the sequence of color patterns R = ( R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R ,R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R ) ..