The Packing Coloring of Distance Graphs D(k,t)
aa r X i v : . [ m a t h . C O ] F e b The Packing Coloring of Distance Graphs D ( k, t ) Jan Ekstein ∗ Pˇremysl Holub ∗ Olivier Togni † June 19, 2018
Abstract
The packing chromatic number χ ρ ( G ) of a graph G is the smallestinteger p such that vertices of G can be partitioned into disjoint classes X , ..., X p where vertices in X i have pairwise distance greater than i .For k < t we study the packing chromatic number of infinite distancegraphs D ( k, t ), i.e. graphs with the set Z of integers as vertex setand in which two distinct vertices i, j ∈ Z are adjacent if and only if | i − j | ∈ { k, t } .We generalize results by Ekstein et al. for graphs D (1 , t ). Forsufficiently large t we prove that χ ρ ( D ( k, t )) ≤
30 for both k , t odd,and that χ ρ ( D ( k, t )) ≤
56 for exactly one of k , t odd. We also givesome upper and lower bounds for χ ρ ( D ( k, t )) with small k and t . Keywords: distance graph; packing coloring; packing chromaticnumber
AMS Subject Classification (2010):
The concept of a packing coloring was introduced by Goddard et al. [9] underthe name broadcast coloring where an application to frequency assignments ∗ University of West Bohemia, Pilsen, Czech Republic,e-mail: { ekstein, holubpre } @kma.zcu.cz . † University of Bourgogne, Dijon, France,e-mail:
[email protected] . packing coloring which we follow in this paper.In this paper we consider simple undirected graphs only. For terminologyand notations not defined here we refer to [1]. Let G be a connected graphand let dist G ( u, v ) denote the distance between vertices u and v in G . We askfor a partition of the vertex set of G into disjoint classes X , ..., X p accordingto the following constraints. Each color class X i should be an i -packing ,a set of vertices with property that any distinct pair u, v ∈ X i satisfiesdist G ( u, v ) > i . Such a partition is called a packing p -coloring , even thoughit is allowed that some sets X i may be empty. The smallest integer p forwhich there exists a packing p -coloring of G is called the packing chromaticnumber of G and it is denoted χ ρ ( G ). The determination of the packingchromatic number is computationally difficult. It was shown to be N P -complete for general graphs in [9]. Fiala and Golovach [7] showed that theproblem remains
N P -complete even for trees.Let D = { d , d , ..., d k } , where d i ( i = 1 , , ..., k ) are positive integers suchthat d < d < ... < d k . The (infinite) distance graph D ( d , d , ..., d k ) has theset Z of integers as a vertex set and in which two distinct vertices i, j ∈ Z areadjacent if and only if | i − j | ∈ D . The study of a coloring of distance graphswas initiated by Eggleton et al. [6] and a lot of papers concerning this topichave been published (see [3], [11], [12], [13], [15] for a sample of results).The study of a packing coloring of distance graphs was initiated byTogni. In [14] Togni showed that χ ρ ( D (1 , t )) ≤
40 for odd t ≥
447 andthat χ ρ ( D (1 , t )) ≤
81 for even t ≥ χ ρ ( D (1 , t )) ≤
35 for odd t ≥
575 andthat χ ρ ( D (1 , t )) ≤
56 for even t ≥ D (1 , t ), for t ≥
9, is also given in [5].In this paper we generalize mentioned results as follows.2 heorem 1.
Let k, t be odd positive integers such that t ≥ and k, t coprime (i. e. D ( k, t ) is a connected distance graph). Then χ ρ ( D ( k, t )) ≤ . Note that, for k = 1, Theorem 1 also improves the upper bound givenin [5]. Corollary 2.
For any odd positive integer t ≥ , χ ρ ( D (1 , t )) ≤ . Theorem 3.
Let k, t be positive integers such that k is odd, t ≥ is evenand k, t coprime (i.e. D ( k, t ) is a connected distance graph). Then χ ρ ( D ( k, t )) ≤ . Theorem 4.
Let k, t be positive integers such that k is even, t ≥ is oddand k, t coprime (i. e. D ( k, t ) is a connected distance graph). Then χ ρ ( D ( k, t )) ≤ . Theorem 5.
Let D ( k, t ) be a connected distance graph, t ≥ . Then χ ρ ( D ( k, t )) ≥ . For k , t both even and also for k, t commensurable, the distance graph D ( k, t ) is disconnected and contains copies of a distance graph D ( k ′ , t ′ ) asits components with k ′ < k , t ′ < t , at least one of k ′ , t ′ odd and k ′ , t ′ coprime(as will be shown in a proof of Lemma 8). In the view of this fact, we cancolor each copy of D ( k ′ , t ′ ) in the same way, thus we obtain the followingstatement. Proposition 6.
Let k, t be positive integers and g their greatest commondivisor. Then χ ρ ( D ( k, t )) = χ ρ (cid:16) D ( kg , tg ) (cid:17) . For small values of k, t we give lower and upper bounds as it is shown inthe following Table 1. Note that, for k = 1, an analogous table was publishedin [5].Throughout the rest of the paper by a coloring we mean a packingcoloring. 3 (cid:31) t 3 4 5 6 7 8 9 102 D (1 ,
2) 14 − D (1 ,
3) 15 − D (1 ,
4) 12 − D (1 , − D (1 ,
2) 13 −
17 14 − D (1 ,
3) 13 −
294 — — 13 − D (2 ,
3) 16 − D (1 ,
2) 15 − D (2 , −
29 13 −
20 14 −
32 13 − D (1 , − D (3 , D (2 , D (3 , −
34 12 −
23 12 −
408 — — — — — — 12 − D (4 , − Table 1: Values and bounds of χ ρ ( D ( k, t )) for 2 ≤ k < t ≤
10. The empha-sized numbers are exact values and all pairs of values are lower and upperbounds. χ ρ ( D ( k, t )) for small k , t In this section we determine new lower and upper bounds for the packingchromatic number of D ( k, t ) which are mentioned in Table 1.For the upper bounds, we found and verified (with a help of a computer)patterns, which can be periodically repeated for a whole distance graph D ( k, t ). This means that we color vertices 1 , . . . , l of D ( k, t ) using a patternof length l and copy this pattern on vertices 1 + pl, . . . , l + pl , p ∈ Z . As mostof these patterns are of big lengths, they do not appear in this paper, butcan be viewed at the web page http://le2i.cnrs.fr/o.togni/packdist/ and tested using the java applet provider. k, t c p Configurations Time2 , . ·
46 hours2 , . ·
327 hours3 , . ·
297 hours3 , . ·
35 hours3 , ·
179 hours4 , ·
23 hoursTable 2: Computations for finding lower bounds of χ ρ ( D ( k, t )). Time of thecomputation is measured on a one-core workstation from year 2012.4 , t q b Configurations Time2 , /
45 32 . ·
14 min3 , /
42 1 . · ,
10 8 47 /
50 10
125 hours4 , /
50 7 . ·
58 hours4 , /
52 1 . ·
145 hours5 , /
50 1 . · , /
50 2 . ·
19 hours5 , /
45 7 . ·
58 hours5 , /
52 1 . ·
107 hours6 , /
50 3 . ·
27 hours7 , /
55 1 . ·
120 hoursTable 3: Computations for finding lower bounds of χ ρ ( D ( k, t )). Time of thecomputation is measured on a one-core workstation from year 2012.For the lower bounds, we followed methods used in proofs of Lemmas6 and 8 in [5]. Some of the lower bounds were obtained using brute forcesearch programs (one in Pascal and one in C++). We showed that a subgraph D p ( k, t ) of D ( k, t ) induced by vertices { , , . . . , p } cannot be colored usingcolors from 1 to c , for some p and c (the results of computations can be seenin Table 2), which implies that χ ρ ( D ( k, t )) ≥ c + 1. For a shortening of acomputation time the programs precolored vertex 1 with color c and tried toextend the coloring for whole D p ( k, t ).For the remaining lower bounds we used a density method. A density ofa color class X i in a packing coloring of G can be defined as a fraction ofall vertices of X i and all vertices of G . For the exact definition of densities d ( X i ) and d ( X ∪ · · · ∪ X i ) we refer to [5].The density method is based on the following proposition. Lemma 7. [8]
For every finite packing coloring with k classes X , X , . . . , X k of a graph G and any positive integer l satisfying ≤ l ≤ k , it holds that k X i =1 d ( X i ) ≥ d ( X ∪ . . . ∪ X l ) + k X i = l +1 d ( X i ) ≥ d ( X ∪ . . . ∪ X k ) = 1 . d ( X ∪ X ∪ · · · ∪ X q ) is bounded by b , for some q and b (the results are summarized in Table 3). For instance, for k, t = 2 , d (1 , , , ≤ /
41. Since there is no pair of vertices in D i − (2 ,
7) with distance greater than i ( i ≥ d ( i ) = 1 / (7 i −
13) for i ≥ d (1 , , , d (5)+ · · · + d (13) ≤ . < χ ρ ( D (2 , ≥
14. The bounds for the other values of k, t in Table 3 areproved similarly, observing that for D ( k, t ), d ( i ) = 1 / ( ti − α ), with α = 13 for k, t = 2 , i ≥ α = 17 for k, t = 3 , i ≥ α = 22 for k, t = 3 , i ≥ α = 8 for k, t = 4 , i ≥ α = 21 for k, t = 4 , i ≥ α = 5 for k, t = 5 , i ≥ α = 10 for k, t = 5 , i ≥ α = 19 for k, t = 5 , i ≥
8; and α = − k = t − i ≥ t − D (2 , D (7 , D (7 , D (8 ,
9) and D (9 ,
10) areobtained from Theorem 5.
First of all we prove for which k , t a distance graph D ( k, t ) is connected. Lemma 8.
A distance graph D ( k, t ) is connected if and only if the greatestcommon divisor of k , t is 1.Proof. If the greatest common divisor of k , t is 1, then from linear algebra1 = mk + nt , where m, n ∈ Z . For every vertex corresponding to a number p ∈ Z it holds that p = pmk + pnt and therefore there exists a path betweenthis vertex and vertex corresponding to 0. Hence it follows that there existsa path between any pair of vertices and that D ( k, t ) is connected.If the greatest common divisor of k , t is p >
1, than we divide a set ofvertices of D ( k, t ) into subsets in terms of equivalence classes modulo p andclearly there is no edge between any pair of vertices from distinct subsetsof vertices of D ( k, t ). Hence D ( k, t ) is not connected which completes theproof. Moreover, vertices corresponding to the equivalence classes modulo p induce p isomorphic copies of a graph D ( kp , tp ) where kp , tp are coprime positiveintegers.A key observation of this section is that a connected distance graph D ( k, t ) can be drawn as k vertex disjoint infinite spirals with t lines or-thogonal to the spirals (e.g. D (3 ,
8) on Fig. 1).6 ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................ 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Figure 1: Distance graph D(3, 8).For i ∈ { , , ..., t − } , the i -band in a connected distance graph D ( k, t ),denoted by B i , is an infinite path in D ( k, t ) on the vertices V ( B i ) = { ik + jt, j ∈ Z } . For i ∈ { , , ..., t − } , the i -strip in a connected distance graph D ( k, t ), t >
24, denoted by S i , is a subgraph of D ( k, t ) induced by the unionof vertices of B i , B i +1 , ..., B i +23 .For a connected graph D ( k, t ) we use a notation D ( k, t ) = S B S B . . . to express that we view D ( k, t ) as a union of strips S , S , . . . and bands B , B , . . . (including edges between strips and bands).It is obvious that an i -strip in D ( k, t ) is isomorphic to an i -strip in D (1 , t ).Hence we apply results in [5] for a coloring of strips in D ( k, t ). Moreover wecolor vertices of all strips using the pattern on 24 ×
24 vertices made by Holuband Soukal in [10], in which it is possible to replace color 16 (17) by 22 (23),because two vertices colored with 16 (17) are in a whole strip at distance atleast 24, respectively (see Fig. 2).
Lemma 9.
Let D ( k, t ) be a connected distance graph, t > , and S i its i -strip. Then it is possible to color S i using colors C = { , , ..., , , , } . Then we can use colors 16 and 17 for coloring of bands as it is explainedin the following statement. 7
Figure 2: A modified pattern on 24 ×
24 vertices.
Lemma 10.
Let D ( k, t ) be a connected distance graph, t ≥ , and B i , B i +25 its bands. Then it is possible to color B i and B i +25 using colors C = { , , , ..., , , , ..., } .Proof. We prove this lemma by exhibiting a repeating pattern using colors 1,16, 17, ..., 21, 24, 25, ..., 30. The pattern was found with help of a computer,it has period 144 and is given here:
8e color B i cyclically with pattern ( ∗ ) starting at the vertex i and B i +25 cyclically with pattern ( ∗ ) starting at the vertex ( i + 25) k + jt for any j ∈{− , − , ..., − , , , ..., } . Let D c be a minimal distance between twovertices colored with the same color c in the same band. Then from thecoloring of a whole band with pattern ( ∗ ) we have D = 26, D = 32, D = 30, D = 32, D = 32 and D = 36. Let u ∈ V ( B i ) and v ∈ V ( B i +25 ) be colored with the same color c . For c ≤
24, the distance between u and v is greater than 24. For c ≥
25, the distance between u and v isat least min {| j | , D c − | j |} + 25 which is greater than c . Hence we have apacking coloring of B i and B i +25 .In proofs of Theorems 3 and 4 we use the following statement proved byGoddard et al. in [9]. Proposition 11. [9]
For every l ∈ N , the infinite path can be colored withcolors l, l + 1 , ..., l + 2 . Now we are ready to prove Theorems 1, 3 and 4.
Proof.
Let D p = {− , − , ..., − , , , ..., } . Let k = min { k (mod 24) , − k (mod 24) } .Let r , s be positive integers such that t = 24 s + r , where r is odd (since t is odd) and minimal such that k ≤ r ≤
33. We prove Theorem 1 evenfor t ≥ r + r , which is in the worst case (for r = 33) the general bound t ≥ s ≥ r and we have s disjoint strips and r disjoint bands suchthat D ( k, t ) = S B S B ...S r − r − B r + r − S r + r ...S s − r .For odd i = 1 , , ..., r − k + 1, we color the strips S i − i − cyclicallywith the pattern from Fig. 2 starting at the vertex 24( i − k + ( i − k . If r > k , then, for even i = 2 , , ..., r − k , we color S i − i − cyclically withthe pattern from Fig. 2 starting at the vertex 24( i − k + ( i − k − t .Let k ≡ k (mod 24). Then, for i = r + 1 , r + 2 , ..., s , we color S i − r cyclically with the pattern from Fig. 2 starting at the vertex 24( i − k + rk − k t . If k = 1, then, for i = r − k + 2 , r − k + 3 , ..., r , we color the strips9 i − i − cyclically with the pattern from Fig. 2 starting at the vertex24( i − k + ( i − k − ( i − r + k − t .Let k ≡ − k (mod 24). Then, for i = r +1 , r +2 , ..., s , we color S i − r cyclically with the pattern from Fig. 2 starting at the vertex 24( i − k + rk + k t . If k = 1, then, for i = r − k + 2 , r − k + 3 , ..., r , we color the strips S i − i − cyclically with the pattern from Fig. 2 starting at the vertex24( i − k + ( i − k + ( i − r + k − t . Hence we have a packing coloringof all s disjoint strips of D ( k, t ) using Lemma 9.For i = 1 , , ..., r , we color the bands B i + i − cyclically with pattern ( ∗ )starting at the vertex 24 ik + ( i − k + j i t such that j i is even (odd) for odd(even) i , respectively, for i > j i − j i − ∈ D p , and for s = r , | j r − j + k | (mod 144) ∈ D p .If s > r , then j i exist. Now assume that r = s .If r = s = 1, then we set j = 0. Note that from k ≤ r we have onlydistance graphs D (1 ,
25) and D (23 , r = s = 5, then we have distance graphs D ( k, k ≤ k ≤ r = 3. For k = 1 , , , , , , , ,
51, we set j = 0, j = − j = −
26 and j , j such that j − j ∈ D p and j = j . For k =21 , , , , , , , ,
71, we set j = 0, j = − j = −
46 and j , j such that j − j ∈ D p and j = j . For k = 73 , , , , , , , ,
123 we set j = 0, j = − j = − j = −
75 and j = −
98. For k = 93 , , , ,
101 we set j = 0, j = − j = − j = −
63 and j = − r = s = 3, then we have distance graphs D ( k,
75) with k ≤
73 suchthat k ≤ r = 3 and we proceed for feasible k in a similar way as in the case r = s = 5.If r = s ≥
7, then we proceed analogously to previous cases (a combina-tion of r numbers from D p could be from 0 to 144, which is the length of thepattern ( ∗ )).Hence we have a packing coloring of all r disjoint bands of D ( k, t ) usingthe same principle as in the proof of Lemma 10.Note that the bands are colored with colors 1 , , , ..., , , , ..., , , ..., , ,
23 in such a way that nopair of adjacent vertices is colored with color 1. Then we conclude that wehave a packing coloring of D ( k, t ), hence χ ρ ( D ( k, t )) ≤ t for which Theorem 1 is true, aregiven in Table 4. r k , , , , , , , , , , ∀ ∀ ∀ ∀ t ≥
25 75 125 175 225 275 325 375 425 r
19 21 23 25 27 29 31 33 k ∀ ∀ ∀ , , , ,
11 5 , , ,
11 7 , ,
11 9 ,
11 11 t ≥
475 525 575 625 675 725 775 825Table 4: Table for t depending on odd k , r with r ≥ k . Proof.
Let D p = {− , − , ..., − , , , ..., } . Let k = min { k (mod 24) , − k (mod 24) } .Let r , s be positive integers such that t = 24( s + 2) + r , where r is even(since t is even) and minimal such that k < r ≤
34. We prove Theorem 3even for t ≥ r + 2) + r , which is in the worst case (for r = 34) the generalbound t ≥ s ≥ r and we have s + 2 disjoint strips and r dis-joint bands such that D ( k, t ) = S S B S ...S r − r − B r + r − S r + r − S r +1)+ r − ...S s +1)+ r − B s +2)+ r − .We color the strip S cyclically with the pattern from Fig. 2 startingat the vertex 0. For odd i = 1 , , ..., r − k , we color the strips S i + i − cyclically with the pattern from Fig. 2 starting at the vertex 24 ik + ( i − k .If r > k + 1, then, for even i = 2 , , ..., r − k −
1, we color S i + i − cyclicallywith the pattern from Fig. 2 starting at the vertex 24 ik + ( i − k − t .Let k ≡ k (mod 24). Then, for i = r + 1 , ..., s + 1, we color S i + r − cyclically with the pattern from Fig. 2 starting at the vertex 24 ik + ( r − k − k t . If k = 1, then, for i = r − k + 1 , r − k + 2 , ..., r , we color thestrips S i + i − cyclically with the pattern from Fig. 2 starting at the vertex24 ik + ( i − k − ( i − r + k ) t .Let k ≡ − k (mod 24). Then, for i = r + 1 , r + 2 , ..., s + 1, wecolor S i + r − cyclically with the pattern from Fig. 2 starting at the vertex114 ik + ( r − k + k t . If k = 1, then, for i = r − k + 1 , r − k + 2 , ..., r , wecolor the strips S i + i − cyclically with the pattern from Fig. 2 starting atthe vertex 24 ik + ( i − k + ( i − r + k ) t . Hence we have a packing coloringof all s disjoint strips of D ( k, t ) using Lemma 9.For i = 1 , , ..., r −
1, we color the bands B i +1)+ i − cyclically withpattern ( ∗ ) starting at the vertex 24( i + 1) k + ( i − k + j i t such that j i iseven (odd) for odd (even) i , respectively, and for i > j i − j i − ∈ D p .We color B s +2)+ r − with a sequence of colors 18, 19, ..., 21, 16, 17,24, 25, ..., 56 starting at any vertex of B s +2)+ r − . By Proposition 11 for l = 18, we can color B s +2)+ r − with colors 18, 19, ..., 56. Since we usedcolors 22 and 23 for a coloring of strips, we replace color 22 (23) by 16(17), respectively, in the coloring of this band. Note the band B s +2)+ r − is the only one with colors greater than 48. By Lemma 10 and the fact thata distance between any vertex of B s +2)+ r − and any vertex of any otherband of D ( k, t ) is at least 49 the defined coloring is a packing coloring of all r disjoint bands of D ( k, t ).Note that the bands are colored with colors 1 , , , ..., , , , ..., , , ..., , ,
23 in such a way that nopair of adjacent vertices is colored with color 1. Then we conclude that wehave a packing coloring of D ( k, t ), hence χ ρ ( D ( k, t )) ≤ t for which Theorem 3 is true, aresummarized in Table 5. r k , , , , , , , , , , ∀ ∀ ∀ ∀ t ≥
98 148 198 248 298 348 398 448 498 r
20 22 24 26 28 30 32 34 k ∀ ∀ ∀ , , , ,
11 5 , , ,
11 7 , ,
11 9 ,
11 11 t ≥
548 598 648 698 748 798 848 898Table 5: Table for t depending on odd k and even r with r > k .12 .3 Proof of Theorem 4 Proof.
Let D p = {− , − , ..., − , , , ..., } . Let k = min { k (mod 24) , − k (mod 24) } .Let s be a positive integer such that t = 24 s + 1 and k = 0. Hence wehave s disjoint strips and 1 band such that D ( k, t ) = S S ...S s − B s . For i = 0 , ...s − S i cyclically with the pattern from Fig. 2starting at the vertex 24 ik . We color B s with a sequence of colors 18, 19, ...,21, 16, 17, 24, 25, ..., 56 starting at any vertex of B s . Hence the band B s is colored with colors 1 , , , ..., , , , ...,
56 and the strips are coloredwith colors 1 , , ..., , ,
23 in such a way that no pair of adjacent verticesis colored with color 1. Then we conclude that we have a packing coloring of D ( k, t ), hence χ ρ ( D ( k, t )) ≤ r , s be positive integers such that t = 24( s + 2) + r , where r isodd (since t is odd) and minimal such that k < r ≤
35. We excludethe previous case k = 0, r = 1 and we prove Theorem 4 even for t ≥ r + 2) + r , which is in the worst case (for r = 35) the general bound t ≥ s ≥ r and we have s + 2 disjoint strips and r disjoint bandssuch that D ( k, t ) = S S B S ...S r − r − B r + r − S r + r − S r +1)+ r − ...S s +1)+ r − B s +2)+ r − .The rest of the proof of Theorem 4 is exactly same as of the proof ofTheorem 3.Some cases, in which we can decrease t for which Theorem 4 is true, aregiven in Table 6. r k , , , , , , , , , , , , , , , ∀ ∀ ∀ ∀ t ≥
25 123 173 223 273 323 373 423 473 523 r
21 23 25 27 29 31 33 35 k ∀ ∀ , , , , ,
12 4 , , , ,
12 6 , , ,
12 8 , ,
12 10 ,
12 12 t ≥
573 623 673 723 773 823 873 923
Table 6: Table for t depending on even k and odd r with r > k .13 .4 Proof of Theorem 5 Proof.
It is shown in [4] that a finite square lattice 15 × D ( k, t ) contains a finite square grid as a subgraphand t ≥ × D ( k, t ).Therefore, χ ρ ( D ( k, t )) ≥
12 for every t ≥ The access to the METACentrum computing facilities, provided under theprogramme ”Projects of Large Infrastructure for Research, Development andInnovations” LM2010005 funded by the Ministry of Education, Youth andSports of the Czech Republic, is highly appreciated.This work was supported by the European Regional Development Fund(ERDF), project NTIS - New Technologies for Information Society, EuropeanCentre of Excellence, CZ.1.05/1.1.00/02.0090.First two authors were supported by the Center of Excellence – Inst. forTheor. Comp. Sci., Prague (project P202/12/G061 of GA ˇCR).The third author was supported by the University of Burgundy (projectBQR 036, 2011).
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