Distance graphs and T-coloring

Distance Graphs and T-Coloring

Gerard J. Chang*

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan

E-mail: gjchangmath.nctu.edu.tw

Daphne D.-F. Liu

-Department of Mathematics and Computer Science, California State University, Los Angeles, Los Angeles, California 90032

E-mail: dliucalstatela.edu

and Xuding Zhu

Department of Applied Mathematics, National Sun Yat-sen University, Kaoshing 80424, Taiwan

E-mail: zhumath.nsysu.edu.tw Received April 3, 1997

We discuss relationships among T-colorings of graphs and chromatic numbers, fractional chromatic numbers, and circular chromatic numbers of distance graphs. We first prove that for any finite integral set T that contains 0, the asymptotic T-coloring ratio R(T ) is equal to the fractional chromatic number of the distance graph G(Z, D), where D=T&[0]. This fact is then used to study the distance graphs with distance sets of the form Dm, k=[1, 2, ..., m]&[k]. The chromatic

numbers and the fractional chromatic numbers of G(Z, Dm, k) are determined for all

values of m and k. Furthermore, circular chromatic numbers of G(Z, Dm, k) for

some special values of m and k are obtained.  1999 Academic Press

1. INTRODUCTION

The T-coloring problem was formulated by Hale [18] as a model for the channel assignment problem, in which an integer broadcast channel is

Article ID jctb.1998.1881, available online at http:www.idealibrary.com on

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Copyright  1999 by Academic Press All rights of reproduction in any form reserved.

* Supported in part by the National Science Council under Grant NSC86-2115-M009-002.

-_{Supported in part by the National Science Foundation under Grant DMS-9805945.} _{Supported in part by the National Science Council under Grant NSC86-2115-M-110-004.}

assigned to each of several locations so that interference among nearby locations is avoided. Interference is modeled by a non-negative integral set T containing 0 (called a T-set) as forbidden channel separations. One can construct a graph G=(V, E) such that each vertex represents a location; two vertices are adjacent if their corresponding locations are nearby. There-after a valid channel assignment or T-coloring is a mapping f from the vertex set of V of G to the set of non-negative integers [0, 1, 2, ...] such that | f (x)& f ( y)| Â T whenever xy # E. The span of a T-coloring f is the difference between the largest and the smallest numbers in f (V), i.e., max[ | f (u)& f (v)|: u, v # V]. Given T and G, the T-span of G, denoted by spT(G), is the minimum span among all T-colorings of G.

T-coloring has been extensively studied in the literature (see [4, 5, 16, 2326, 3032, 35]). Let _ndenote spT(Kn), where Kn is a complete graph

with n vertices. Griggs and Liu [16] proved that the difference optimum sequence, 2_=(_n+1&_n)n=1, is eventually periodic. This implies that for

any T-set, the asymptotic T-coloring ratio R(T ) := lim

nÄ 

_n

n

exists and is a rational number. This result was also proven by Rabinowitz and Proulx [30] and Cantor and Gordon [1] by different approaches.

The notion of distance graphs originated with the plane-coloring problem: What is the smallest number of colors needed to color all points of a Euclidean plane such that points at unit distances are colored with different colors. It is well known that four colors are necessary [28] and seven colors are sufficient [17]. However, the exact number of colors needed remains unknown (see [6]). Motivated by this problem, Eggleton [10] made the following generalization. Suppose S is a subset of a metric space M with metic d, and D is a set of positive real numbers. The distance graph G(S, D) with distance set D is the graph with vertex set S and edge set [xy: d(x, y) # D]. The objective is to determine /(S, D), the chromatic number of G(S, D). Note that the plane-coloring problem introduced above is equivalent to finding /(R2_{, [1]).}

Let D be a set of positive integers (called a D-set). The distance graph to be studied in this article is G(Z, D), which has Z as the vertex set and [uv: |u&v| # D] as the edge set. The problem of finding /(Z, D) for different D-sets has been studied extensively (see [2, 7, 9, 1114, 21, 3741]). A fractional coloring of a graph G is a mapping c from I(G), the set of all independent sets of G, to the interval [0, 1] such that x # I # I(G)c(I )1 for all vertices x in G. The fractional chromatic number

/f(G) of G is the infimum of the value I # I(G)c(I ) of a fractional coloring

For a given T-set, letting D=T&[0], Liu [26] proved that the asymptotic T-coloring ratio R(T ) is a lower bound of /(Z, D). Hence, T-colorings and distance graphs are closely related. We shall explore further relationships between these two concepts. In Section 2, we prove that for any T-set, R(T ) is equal to the fractional chromatic number of the distance graph G(Z, D), if D=T&[0]. This relationship provides new insights concerning the parameter R(T ), and can be used to simplify the proofs of some known results regarding R(T ).

Section 3 focuses on the family of distance graphs with D-sets of the form D_{m, k}=[1, 2, ..., m]&[k]. The chromatic numbers of such distance graphs, denoted as /(Z, Dm, k), have been investigated in the following articles.

In [11], Eggleton, Erdos and Skilton obtained the solution for k=1, and partial solutions for k=2: /(Z, Dm, 1)=w(m+3)2x for any m2,

/(Z, Dm, 2)=w(m+4)2x when m3 (mod 4), and w(m+3)2x

/(Z, Dm, 2)w(m+5)2x for any m4 with m#3 (mod 4). For 3k<m,

the same authors provided the bounds

max

{

k,

\

1 2

\

m k&1+1

+

t

=

/(Z, Dm, k)min

{

m,

\

1 2

\

m k+3

+

k

=

, where t=2 if k=3, and t=k&2 if k4. The same result for the case k=1 was also proven by Kemnitz and Kolberg in [21] by a different approach.

The lower bound of /(Z, Dm, k) in the above has been improved to

W(m+k+1)2X by Liu ([26]), who also showed that the new bound is sharp for all pairs of integers (m, k) where k is odd. Furthermore, complete solutions for k=2 and 4, and partial solutions for other even integers k are given in [26].

The main results of this paper are complete solutions for the chromatic numbers and the fractional chromatic numbers of the distance graphs G(Z, D_{m, k}) for all values m and k. These results are also applied to study the circular chromatic number of distance graphs.

Suppose k and d are positive integers such that k2d. A (k, d )-coloring of a graph G=(V, E) is a mapping c from V to [0, 1, ..., k&1] such that &c(x)&c( y)&kd for any edge xy in G, where &a&k=min[a, k&a]. The

circular chromatic number /c(G) of G is the infimum of kd for all (k, d

)-colorings of G. The circular chromatic number is also known as the star-chromatic number in the literature (see [36, 42, 43]).

For any graph G, it is well known that

max

{

|(G),|V(G)|

The parameters involved in (V) for distance graphs are explored in this paper. For simplicity, let |(Z, D), :(Z, D), /f(Z, D), and /c(Z, D) denote

the clique number, the independence number, the fractional chromatic number, and the circular chromatic number of G(Z, D), respectively.

2. RELATIONSHIPS BETWEEN R(T ) AND /f(Z, D)

This section shows that for any T-set, the asymptotic T-coloring ratio R(T ) is equal to the fractional chromatic number of the distance graph G(Z, D), if D=T&[0]. Based upon this result, we give simpler and different proofs of some known facts regarding R(T ).

Theorem 1. For any finite T-set, if D=T&[0], then R(T )=/

f(Z, D).

Proof. Suppose c is an optimal T-coloring of Kn, where 0=c(1)<

c(2)< } } } <c(n)=_n. Let m=1+maxd # Dd. For 1ic(n)+m, let

I_i=[ j # Z : j#i+c(k) (mod c(n)+m) for some k with 1kn]. It is straightforward to verify that each Iiis an independent set in G(Z, D),

and that every integer belongs to exactly n of the independent sets Ii

(1ic(n)+m). Define a mapping c$: I(G(Z, D)) Ä [0, 1] as

c$(I)=

{

1n, if I=Ii for 1ic(n)+m;

0, otherwise.

Then c$ is a fractional coloring of G(Z, D). This implies that for any positive integer n, /f(Z, D)(c(n)+m)n. Hence, we have

/_f(Z, D) lim nÄ  c(n)+m n = limnÄ  _n n=R(T ).

To show /f(Z, D)R(T ), let Gnbe the subgraph of G(Z, D) induced by

the vertex set [0, 1, 2, ..., c(n)]; i.e., Gn=G([0, 1, 2, ..., c(n)], D). Then the

set of vertices [c(1), c(2), ..., c(n)] is a maximum independent set in Gn. So,

for any positive integer n, we have /_f(Z, D)/f(Gn)

|V(Gn)|

:(Gn)

=c(n)+1

This implies /f(Z, D) lim nÄ  c(n)+1 n = limnÄ  _n n=R(T ). Therefore, R(T )=/f(Z, D). Q.E.D

The theorem above provides new insights into the asymptotic T-coloring ratio R(T ). Some previous results concerning R(T ) can be obtained from this approach. For example, it is well known that R(T )2, provided T{[0] (see [1, 16, 30]). This is straightforward when we consider fractional chromatic numbers, since the fractional chromatic number of any non-trivial graph is at least 2. Moreover, for a non-trivial graph G, /f(G)=2 if and only if G is bipartite. Since G(Z, D) is bipartite if and only

if D contains no even integers (assuming that gcd(T )=1), we have the following result.

Corollary 2. For any T-set with gcd(T )=1, R(T )=2 if and only if T contains only odd integers except 0.

Theorem 1 can also be applied to some other known results about R(T ) which are closely related to an earlier number theory problem, namely, sequences with missing differences. Given a T-set, a T-sequence is an increasing sequence S of nonnegative integers such that x& y  T for any x, y # S. Motzkin [29] proposed studying the supremum +(T ) of the asymptotic upper densities of these sequences S. Cantor and Gordon [1] determined the exact values of +(T ) when |T | =2 and 3. Haralambis [19] gave partial solutions when |T | =4 or 5. It is known that +(T ) is equal to the reciprocal of R(T ) (see [16]). Therefore results on sequences with missing differences can be applied to T-colorings. Cantor and Gordon [1] and Rabinowitz and Proulx [30] proved that if T=[0, a, b], gcd(a, b)=1, and a and b are of different parity, then R(T )=2(a+b)(a+b&1). The original argument in proving the inequality R(T )2(a+b)(a+b&1) in [1] was quite complicated. However, applying some facts about fractional chromatic number, one can obtain the following simpler proof. Since a and b are of opposite parity, G(Z, D) with D=[a, b] contains an odd cycle Ca+b. As /f(C2m+1)=2+1m, it follows that /f(Z, D)/f(Ca+b)=

2(a+b)(a+b&1). Thus, by Theorem 1, R(T )2(a+b)(a+b&1). Indeed, combining Theorem 1 with the fact that /f(H)/f(G) if H is a

subgraph of G, the following is obvious.

Corollary 3. If H is a subgraph of the distance graph G(Z, D), then /f(H)R(T ), where T=D _ [0].

3. VALUES OF /f(Z, Dm, k), /(Z, Dm, k), AND /c(Z, Dm, k)

In this section, we first calculate /f(Z, Dm, k), which, according to (V), is

a lower bound of /(Z, Dm, k). We then determine /(Z, Dm, k) for all values

of m and k. Using this approach, circular chromatic numbers of G(Z, Dm, k)

for special values of m and k are obtained as well.

The values of R(T ) as T=Dm, k_ [0] are given in [26]. Thus, the

following two results can also be obtained by using Theorem 1. Here we include methods of calculating /f(Z, Dm, k) directly.

Theorem 4. If 2k>m, then

|(Z, Dm, k)=/f(Z, Dm, k)=/c(Z, Dm, k)=/(Z, Dm, k)=k.

Proof. Since the set of vertices [1, 2, ..., k] forms a clique in G(Z, Dm, k), k|(Z, Dm, k).

By (V), it is sufficient to show /(Z, Dm, k)k. Define a vertex-coloring f

on Z as f (i)=(i mod k). Then f is a proper coloring, since Dm, k contains

no multiple of k. Q.E.D

Theorem 5. If 2km, then /

f(Z, Dm, k)=(m+k+1)2.

Proof. For any i with 0  i  m + k, Ii= [ j # Z : j & i # 0 or k

(mod m+k+1)] is an independent set in G(Z, Dm, k). Furthermore, each

integer is contained in exactly two such independent sets. Define a mapping c: I(G(Z, Dm, k)) Ä [0, 1] as

c(I )=

{

1 2,

0,

if I=Iifor 0im+k;

otherwise.

It is easy to check that c is a fractional coloring of G(Z, Dm, k). Thus,

/_f(Z, Dm, k)(m+k+1)2.

Since :([0, 1, ..., m+k], Dm, k)=2 for 2km, by (V), /f([0, 1, ...,

m+k], Dm, k)(m+k+1)2. Therefore, the proof is complete. Q.E.D

Because /(G) is an integer, Theorem 5 and ( V ) imply the following:

Corollary 6 [26]. If 2km then /(Z, D

m, k)W(m+k+1)2X.

We are now in a position to given the complete solutions to /(Z, Dm, k)

for all values of m and k. This is accomplished in the next two results. As will be shown, /(Z, Dm, k) is either W(m+k+1)2X or W(m+k+1)2X+1.

Lemma 7. Suppose 2km. Write m+k+1=2rm$ and k=2sk$, where r and s are non-negative integers and m$ and k$ are odd integers. If 1rs, then /(Z, Dm, k)>(m+k+1)2.

Proof. Since 1r, m+k+1 is even. Assume to the contrary that

/(Z, D_{m, k})(m+k+1)2. By Corollary 6, /(Z, Dm, k)=(m+k+1)2.

Color G(Z, Dm, k) by using (m+k+1)2 colors.

For each integer i, consider the subgraph of G(Z, Dm, k) induced by the

m+k+1 vertices [i, i+1, ..., i+m+k]. This subgraph has independence number 2. Hence, each of (m+k+1)2 colors is used at most, and hence exactly, twice in this subgraph. Thus, each color is used exactly twice in any consecutive m+k+1 vertices. Consequently, vertices i and i+m+ k+1 have the same colors for all i # Z. Therefore, for each i # S := [0, 1, ..., m+k], the only possible vertices in S having the same color as i are i+k and i&k (mod m+k+1).

Consider the circulant graph C(m+k+1, k), with vertex set S and in which vertex i is adjacent to vertex j if and only if j#i+k or i&k (mod m+k+1). It follows from the discussion in the preceding paragraph that two vertices x and y of S have the same color only if xy is an edge of the circulant graph C(m+k+1, k). Since the intersection of each color class with S contains exactly two vertices, the coloring induces a perfect matching of C(m+k+1, k). However, C(m+k+1, k) is the disjoint union of d cycles of length (m+k+1)d, where d=gcd(m+k+1, k). Since C(m+k+1, k) has a perfect matching, each cycle has an even length. This

implies that r>s, contrary to the assumption rs. Q.E.D

The next theorem determines the chromatic number /(Z, Dm, k) for all

values of m and k. Incidentally, it also shows that the converse of Lemma 7 is true.

Theorem 8. Suppose 2km. Write m+k+1=2rm$ and k=2sk$,

where r and s are non-negative integers and m$ and k$ are odd integers. Then

/(Z, D_{m, k})=

{

m+k+12 ,

\

m+k+22



,

if r>s; otherwise.

Proof. It follows from Corollary 6 and Lemma 7 that if r>s, then

/(Z, Dm, k)(m+k+1)2; if rs, then /(Z, Dm, k)W(m+k+2)2X.

Therefor it suffices to show that G(Z, Dm, k) is (m+k+1)2-colorable, if

r>s; and G(Z, Dm, k) is W(m+k+2)2X-colorable, if rs. It is known that

coloring for the subgraph of G(Z, Dm, k) induced by all non-negative

integers.

We first decompose k into the sum of an odd number of integers, k=a1+a2+ } } } +ap, as follows:

Case 1. For r>s or s=0, let p=k$ and aj=2s for 1 jp.

Case 2. For rs{0, let p=k&1 and aj=1 for 1 j<p and ap=2.

Next, partition the set Z into consecutive blocks of sizes a1, a2, ..., ap

periodically. Then ``pre-color'' the blocks, alternating RED and BLUE. We call a vertex RED (or BLUE) if it falls within a RED (or BLUE) block.

Define a coloring f on the vertices of all non-negative integers of G(Z, D_{m, k}) according to the following three rules:

(R1) f (i)=i, if 0ik&1;

(R2) f (i)= f (i&k), if i is BLUE and ik;

(R3) f (i)= the smallest non-negative integer that has not been used as a color in the m vertices preceding i, if i is RED and ik.

To show that f is a proper coloring, we claim that for any vertex i, f (i){ f ( j) for all j{i&k with i&m j<i. It is easy to see that the claim is true when (R1) or (R3) is performed. Suppose (R2) is executed, i.e., i is BLUE, ik, and f (i)= f (i&k). Since k is divided into an odd number of blocks, i&k is a RED vertex. By (R1) or (R3), f (i&k) is different from any of the colors of the m vertices preceding i&k. Thus, it is sufficient to show that f (i){ f ( j) for all j with i&k< j<i.

If j is RED, by (R1) or (R3), f ( j){ f (i&k) and so f (i){ f ( j). If j is BLUE, by (R1) of (R2), f ( j)= f ( j&k). One has i&k&m<j&k<i&k (because 2km), so f (i&k){ f ( j&k). This implies f (i){ f ( j).

To complete the proof of the theorem, it is sufficient to show that f is an ((m+k+1)2)-coloring if r>s; and f is an W(m+k+2)2X-coloring if rs. One can accomplish this by counting the number of colors that have been used for the m vertices preceding a RED vertex i for which ik. The first k vertices need at most k colors. For the remaining m&k vertices, only those RED vertices need new colors.

If r>s, then m&k+1 is a multiple of 2s+1_{. Any consecutive 2}s+1

vertices have 2s _{BLUE vertices and 2}s _{RED ones, so there are exactly}

(m&k&1)2 RED vertices in the remaining m&k vertices. Therefore, the total number of colors used in f is at most k+(m&k&1)2+1= (m+k+1)2.

If rs (with s=0 in Case 1 and s{0 in Case 2), then there are at most W(m&k)2X RED vertices in the remaining m&k vertices. Thus, the total

number of colors used in f is at most k+W(m&k)2X+1=

We now present the following two results concerning the circular chromatic number of G(Z, Dm, k). The first one follows from (V) and

Theorems 5 and 8.

Corollary 9. Suppose 2km. Write m+k+1=2rm$ and k=2sk$,

where r and s are non-negative integers and m$ and k$ are odd integers. If r>s, then /f(Z, Dm, k)=/c(Z, Dm, k)=/(Z, Dm, k)=(m+k+1)2.

Theorem 10. If 2km and k is relatively prime to m+k+1, then

/f(Z, Dm, k)=/c(Z, Dm, k)=(m+k+1)2.

Proof. Since k is relatively prime to m+k+1, there exists an integer n such that nk#1 (mod m+k+1). Consider the mapping c defined by c(i)=(in mod m+k+1) for all i # Z. For any edge ij in G(Z, D_{m, k}), we shall prove that &c(i)&c( j)&m+k+12. Suppose to the contrary, that

&c(i)&c( j)&m+k+11; i.e., c(i)&c( j)#0 or 1 or &1 (mod m+k+1).

Then i& j#0 or k or &k (mod m+k+1), which contradicts the fact that i is adjacent to j. Thus c is an (m+k+1, 2)-coloring of G(Z, Dm, k). This

along with Theorem 5 and ( V ) implies the theorem. Q.E.D

Remarks. Many new results related to this topic have been obtained since the submission of this paper. In [3], the circular chromatic numbers of all the graphs G(Z, D_{m, k}) are determined. The chromatic number, circular chromatic number and fractional chromatic number of distance graphs with distance sets of the form Dm, k, s=[1, 2, ..., m]&[k, 2k, ..., sk]

have been studied in [8, 20, 27, 44]. (Accordingly, the distance graphs discussed in this paper are G(Z, Dm, k, 1).) In [27], the chromatic numbers

of all the graphs G(Z, Dm, k, 2) are determined. The same paper also

deter-mined the fractional chromatic numbers of all the graphs G(Z, Dm, k, s). In

[8], the following was proved:

W(m+sk+1)(s+1)X/(G(Z, Dm, ks))W(m+sk+1)(s+1)X+1.

Moreover, both the upper bound and the lower bound are attainable. Then in [20], the chromatic numbers of all the graphs G(Z, Dm, k, s) are

com-pletely determined. Finally and most recently, [44] determines the circular chromatic numbers of all the graphs G(Z, Dm, k, s).

ACKNOWLEDGMENT

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