Channel assignment on Cayley graphs

Channel Assignmenton Cayley Graphs

Patrick Bahls

DEPARTMENT OF MATHEMATICSUNIVERSITY OF NORTH CAROLINA, ASHEVILLE, NORTH CAROLINA

E-mail: [email protected]

Received April 7, 2009; Revised April 19, 2010

Published online 31 August 2010 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/jgt.20523

Abstract: We address various channel assignment problems on theCayley graphs of certain groups, computing the frequency spans byapplying group theoretic techniques. In particular, we show that if G is theCayley graph of an n-generated group � with a certain kind of presentation,then �(G;k,1)≤2(k+n−1). For certain values of k this bound gives theobvious optimal value for any 2n-regular graph. A large number of groups(for instance, even Artin groups and a number of Baumslag–Solitar groups)satisfy this condition. � 2010 Wiley Periodicals, Inc. J Graph Theory 67: 169–177, 2011

MSC 2000: 05C78; 05C15; 90B18

Keywords: channel assignment; frequency assignment; L(2,1)-labeling; Cayley graph

1. INTRODUCTION

Recall that channel assignment problems arise from the following setting: imagine thatwe are told the relative positions of a collection {T1,T2, . . . ,Tn} of radio transmitters towhich we must assign channels or frequencies fi = f (Ti) in such a fashion that if twotransmitters are close enough to one another, their frequencies differ by a predetermined

Contract grant sponsor: NSF REU; Contract grant number: DMS-0647804.Journal of Graph Theory� 2010 Wiley Periodicals, Inc.

169

170 JOURNAL OF GRAPH THEORY

amount. We are then asked to find the overall smallest (and therefore most economical)range of frequencies from which a suitable assignment of frequencies may be made.Such problems in the plane were first suggested by Hale in [6]; they have been heavilyinvestigated in the nearly three decades since this work.

This planar problem has a natural graph-theoretical analogue, introduced by Griggsand Yeh in [5]. In the graph setting, the vertices V of a simple graph G representthe transmitters, and the distance between two vertices is measured by the usual pathmetric � induced by the graph’s edge set E. For instance, we may be asked to find thesmallest value � such that there exists a vertex labeling � :V →{0,1, . . . ,�} satisfying|�(u)−�(v)|≥ki if �(u,v)= i, for i=1,2. Such a labeling is called an L(k1,k2)-labeling.An enormous amount of work has been done in computing this span �(G;k1,k2) invarious settings. See [1] for a comprehensive survey of relevant work.

Note. Sometimes �(G;k1,k2) is defined to be the minimum over all valid labelings � :V →N of the difference maxv∈V �(v)−minv∈V �(v). However, without loss of generalitywe may assume that minv∈V �(v)=0, and we make this assumption throughout thesequel.

The present work is motivated in part by the recent work [4] of Griggs and Jin, inwhich the spans �(G;k,1) are computed for the three regular lattices in the Euclideanplane, for real values k. Note that here we consider only integral values for k.

Our attention here is on infinite graphs arising as the undirected graphs underlyingCayley graphs of certain groups. We recall that the Cayley graph of the group � relativeto the finite group generating set S is the labeled directed graph (V ,E) for which V =�and E={(u,us)|u∈V ,s∈S}, where the edge (u,us) is labeled s. That is, there is anedge labeled s between two vertices of G if one is obtained from the other throughright multiplication by s. Note that if |S|=n, then the undirected graph underlying theCayley graph G is 2n-regular if for all s, s′ ∈S, ss′ �=1.

Channel assignment on certain Cayley graphs has been considered before, althoughnot extensively. For instance, in [11, 12, 2] Zhou et al., consider Cayley graphs ofabelian groups. Although some of the groups we examine here are abelian, we obtainbounds on � for many nonabelian groups as well.

We also note that some of the techniques we develop below are similar to those of vanden Heuvel et al. [7], who consider some of the same graphs as we do here. However,our results generalize some of the familiar infinite graphs (the infinite triangular andsquare lattices, for instance) in a different manner than that considered in [7].

Although we will make use of the edge labels and orientations in defining ourlabeling, these data can be ignored once the labeling is complete, allowing us to considerthe underlying undirected graph.

Let S be a set, and let S−1 be the set of formal inverses of elements of S, soS−1 ={s−1|s∈S}. Let R be a set of relations over the alphabet S±1 =S∪S−1. That is,each element r∈R is an equation of the form w1 =w2, where w1 and w2 are words inthe alphabet S±1. We say that the group � has presentation 〈S|R〉 if � is isomorphic tothe quotient group F(S) /N(R), where F(S) is the free group generated by the set S, andN(R) is the smallest normal subgroup of F(S) generated by the relations in R. Roughlyspeaking, an element �∈� is trivial if and only if it can be written as a product ofconjugates of terms (w1w−1

2 )�, �=±1, for w1 =w2 an element of R. Without loss ofgenerality we may assume that all of the relations in R have the form w=1, and wewill make this assumption throughout the remainder of the article.

Journal of Graph Theory DOI 10.1002/jgt

CHANNEL ASSIGNMENT ON CAYLEY GRAPHS 171

Fix a generating set S and let w be a word in S±1. The exponent sum of s in w, denotedexps(w), is the sum of the exponents on occurrences of s in w (including both positiveand negative exponents). For a fixed natural number N ∈N we say that a presentation〈S|R〉 is (s,N)-balanced if for every relation (w=1)∈R, exps(w)≡0 mod N. We saythat the presentation is N-balanced if it is (s,N)-balanced for every s∈S.

Our main result is the following

Theorem 1.1. Let k be a positive integer, S a set with n elements, R a set of relationsover the alphabet S±, and � the group with presentation 〈S|R〉. Suppose that ss′ �=1for all s, s′ ∈S and that the presentation 〈S|R〉 is (2(n+k)−1)-balanced. Then forthe underlying undirected graph G of the Cayley graph G(�,S) we have �(G;k,1)≤2(k+n−1), and equality holds if k≤2n.

Note. The upper bound obtained here is dramatically lower than the upper boundfor general n-regular graphs, as illustrated by the easy example of the complete graphKn+1: �(Kn+1;k,1)=nk.

As we will see in the following section, it follows from [3] that if k≤d then�(G;k,1)≥2k+d−2 for all d-regular simple graphs, finite or infinite. Moreover,Section 2 will close with a slightly more general version of Theorem 1.1 whose proofis entirely analogous.

As a sample application of Theorem 1.1, the integer lattice graph G=Z×Z is the4-regular Cayley graph of the Artin group A=〈s, t|st= ts〉 (see Section 3). We thusrecapture the value �(G;2,1)=6, derived in [4] and elsewhere.

We consider other specific examples in Section 3.

2. PROOF OF THE MAIN THEOREM

We begin this section by noting that the proposed span �(G;k,1)=2(k+n−1) is thebest possible for a 2n-regular simple graph, if k is “small.” The following propositionwas proven by Georges and Mauro [3]:

Proposition 2.1. Let G be a d-regular graph, and let k≤d. Then �(G;k,1)≥2k+d−2.

Notes. In the case k≤2, any Cayley graph will satisfy the condition k≤d, sincek≤2<2n+1 even if G is 1-generated. To see that Proposition 2.1 may fail whenk≥d+1, let d=2� and k=d+1, and consider the d-regular infinite tree G (G isthe Cayley graph of the free group F� on � generators). We label G’s vertices usingthe values A={0, . . . ,d−1} and B={2k−2, . . . ,2k+d−3}. Label V(G) so that eachA-vertex is adjacent to precisely one B-vertex of each type, and analogously for eachB-vertex. This yields a labeling with span 2k+d−3<2k+d−2. For a finite example,let d≥1 be given and consider the complete bipartite graph G=Kd,d, with k=d+1and A and B as above. We label one part of G with A and the other with B; the resultinglabeling has span 2k+d−3 as well.

Until further notice, suppose that � is given by the (2(k+n)−1)-balanced presen-tation 〈S|R〉 with n=|S| and S={s1, . . . ,sn}, and that G=G(�,S) is the corresponding2n-regular Cayley graph. Let m=2(k+n)−1.



We define our labeling first on the set S, by letting �(si)=k+ i−1 for i=1, . . . ,n.Next, extend � to S−1 by letting �(s−1

i )=m−�(si)=m−k− i+1 for i=1, . . . ,n. This� then extends to a group homomorphism from the free group F(S) to Zm. Explicitly,for a word w=s�1

i1. . .s�r

ir(�j =±1 for all j) in the free group,

�(w)=�(s�1i1

. . .s�rir

)=�1�(si1 )+·· ·+�r�(sir )=n∑

j=1expsj

(w)�(sj),

where addition is performed in Zm.

Lemma 2.2. The map � :F(S)→Zm induces a group homomorphism � :�→Zm.

Proof. This is an immediate consequence of the fact that 〈S|R〉 is m-balanced.By Dyck’s Theorem (see [10]), we need only show that every relation r in R is preservedby the action of �. Indeed, let r= (w=1)∈R. Then

�(w)=n∑

j=1expsj

(w)�(sj)≡0=�(1),

where equivalence is modulo m. Thus the relation r is preserved. �

Since V =V(G)=�, � serves as a labeling for V as well. Note that if u, v∈V areadjacent, then without loss of generality v=us for some s∈S, so that

�(v)−�(u)=�(us)−�(u)=�(s)≥k,

so � satisfies the adjacency condition demanded by a L(k,1)-labeling. For the conditionon vertices at distance two from one another, note that if �(u,v)=2, then v=ust forsome s, t∈S±1.

Lemma 2.3. Let s, t∈S±1, s �= t−1. Then �(s)+�(t) �≡0 mod m.

Proof. It is easy to check that if s, t∈S, 0<�(s)+�(t)<m, and if s, t∈S−1, m<

�(s)+�(t)<2m. In either case, �(s)+�(t) �≡0 mod m.Without loss of generality, assume s=si ∈S and t= (sj)−1 ∈S−1. Then �(s)+�(t)=

i+k−1+m− j−k+1=m− j+ i �=m, so �(s)+�(t) �≡0 mod m in this case too. �

Thus if v=ust, �(v)=�(u)+�(s)+�(t) �=�(u), so the labels on u and v are distinct,as desired. Thus � shows that �(G;k,1)≤m−1=2(k+n−1). Combining this withProposition 2.1 proves our main theorem.

Note. It is easy to show that our labelings are often no-hole; that is, all valuesin {0, . . . ,�}, with �=2(n−k+1), are used. Obviously, the value 0 labels the vertexcorresponding to the identity, and the values {k, . . . ,k+2n−1} label the vertices corre-sponding to S±1. Suppose that k≤n. Note that �(s−1

n sr)=2k+n+r−1, so we can let rrun through the set {n−k+1, . . . ,n−1} in order to capture the labels {k+2n, . . . ,2k+2n−2}. Moreover, since �(s−1

1 sr)=2k+2n+r−2 we can let r run through the set{2, . . . ,k} in order to capture the labels {1, . . . ,k−1}. That the vertices s−1

1 sr and s−1n sr′

are distinct for all choices of r and r′ follows easily from the hypothesis that thepresentation is balanced. Thus all labels are used in the case k≤n.

We may relax the condition ss′ �=1 slightly. Let us allow our generating set to containa single involution, an element such that s2 =1. In this case, the corresponding Cayley



graph will be regular of degree d=2n−1, since at each vertex we can identify the“incoming” and “outgoing” edges labeled by the single involution.

Theorem 2.4. Let k∈N be fixed. Let � be the group given by the presentation

〈S∪{t}|R∪{t2 =1}〉,

where |S|=n−1 and R is a (2(k+n−1))-balanced set of relations in (S∪{t})±1.Suppose that ss′ �=1 for all s,s′ ∈S. Let G=G(�, (S∪{t})) be the undirected (2n−1)-regular graph underlying the Cayley graph of � relative to S∪{t}. Then �(G;k,1)≤2(k+n)−3, and equality holds if k≤2n−1.

Note that the purported span 2(k+n)−3=2k+d−2, just as in Theorem 1.1.

Proof. Let m=2(n+k−1) and let S={s1, . . . ,sn−1}. We may let sn = t, so that s2n =1

or sn =s−1n . As before we define � :{s1, . . . ,sn}→{0, . . . ,2(k+n)−3} by �(si)=k+ i−1

and �(s−1i )=m−k− i+1. Note that �(sn)=k+n−1, so �(sn)=�(s−1

n ).Just as before this � extends first to a homomorphism from F(s1, . . . ,sn) to the group

Zm, then to a homomorphism from � to Zm. Once more it is easily verified that �

satisfies the required separation conditions, showing that �(G;k,1)≤2(k+n)−3, andequality is obtained for k≤2n−1 because of Proposition 2.1. �

We note one more variation of the main theorem. Recall that � :V →N is an L(k1,k2)-labeling if �(u,v)= i implies that |�(u)−�(v)|≥ki, for i=1,2. The following is provenin exactly the same way as in the main theorem:

Theorem 2.5. Let k1,k2 ∈N be fixed, and let � be the group given by the (2(k1 +(n−1)k2)+1)-balanced presentation 〈S|R〉, where n=|S|. Suppose that ss′ �=1 for alls,s′ ∈S. Let G=G(�,S) be the undirected 2n-regular graph underlying the Cayleygraph of � relative to S. Then �(G;k1,k2)≤2(k1 +(n−1)k2).

Clearly this reduces to the main theorem when we set k1 =k and k2 =1.

3. SPECIFIC EXAMPLES

The examples in this section are meant to be illustrative and not exhaustive. Indeed, themethod we have developed here applies to an incredibly diverse collection of Cayleygraphs, and in the concluding section we will consider a generalization of the methodwhich addresses almost all Cayley graphs in certain interesting classes. For the timebeing, let us indicate a few of the more important examples of graphs for which ourmethod yields results. In all of the following examples, the equalities in special casesare consequences of Proposition 2.1.

Given two generators s and t of a group G and m∈N, let wm(s, t) denote the alternatingproduct stst · · · of length m. An Artin group is any group given by a presentation of theform 〈S|R〉 where S={s1, . . . ,sn} and

R={wmij(si,sj)=wmij(sj,si)|mij ∈N∪{∞}, mij =mji, and mij =1⇔ i= j}.



We take mij =∞ to mean there is no relation involving si and sj. S is called thefundamental generating set for A. We say that such A is even if every mij is eitherinfinite or even whenever i �= j.

Theorem 1.1 immediately implies the following

Corollary 3.1. Let G=G(A,S) be the Cayley graph of an even Artin group A, withrespect to the fundamental generating set. Then �(G;k,1)≤2(k+|S|−1), and equalityholds if k≤2|S|.

For example, as mentioned in the introduction, the integer lattice graph G=Z×Z is the Cayley graph of A=〈s, t|st= ts〉, and so �(G;k,1)≤2(k+2−1)=2k+2. Inparticular, �(G;2,1)=6, �(G;3,1)=8, and �(G;4,1)=10, recovering part of the formulagiven in Theorem 3.4 of [4]. (That theorem gives a precise value of �(G;k,1)=k+6for k≥5.)

We may generalize Z×Z in a different manner. This graph corresponds to thefundamental group of the genus-1 torus. More generally, the 0-balanced (and thereforeN-balanced, for any N) one-relator presentation

〈s1,s2, . . . ,sn, t1, t2, . . . , tn|s1t1s−11 t−1

1 . . .sntns−1n t−1

n =1〉gives the fundamental group of a genus-n torus, given n∈N. Theorem 1.1 showsthat the Cayley graph G of this group relative to the given generating set has span�(G;k,1)≤2(k+2n−1), with equality obtaining whenever k≤4n.

Another interesting family of groups are the Baumslag–Solitar groups; BS(1,r) isgiven by the presentation 〈a,b|ab=bar〉. These groups arise frequently in applicationsof group theory to geometry and topology, and are canonical examples of highly “non-hyperbolic” groups. The following is an immediate consequence of the main theorem:

Corollary 3.2. Let k be fixed and let r≡1 mod 2k+3. If G is the Cayleygraph of the Baumslag–Solitar group BS(1,r), then �(G;k,1)≤2k+2, with equalityfor k≤4.

As a sample application of Theorem 2.4, consider the infinite ladder L∞, two infinitepaths with corresponding vertices connected by a single edge. L∞ is the Cayley graphcorresponding to the group presentation 〈s, t | st= ts, t2 =1〉. Theorem 2.4 shows that�(L∞;k,1)≤2(k+n)−3=2k+1, with equality whenever k≤3. Thus �(L∞;2,1)=5and �(L∞;3,1)=7.

Note that the span � is “weakly inherited” by subgraphs: if H ≤G, �(H;k,1)≤�(G;k,1). Using this property of � we obtain the following

Corollary 3.3. Let L be any ladder, finite or infinite. Then �(L;k,1)≤2k+1.

By identifying the ends of a ladder we obtain a prism Prn, consisting of two cyclesof length n, corresponding points on which are connected by a single edge. Notethat Prn is the Cayley graph corresponding to the presentation 〈s, t|sn =1, t2 =1〉, withinvolution t. We may now apply Theorem 2.4 to obtain

Corollary 3.4. Let k be fixed. If n≡0 mod 2k+2, then �(Prn;k,1)≤2k+1, andequality holds when k≤3.

For instance, we recover the well-known fact �(Prn;2,1)=5 when n≡0 mod 6.



4. GENERALIZATIONS AND DIRECTIONS FOR FURTHER STUDY

As indicated at the start of the preceding section, this article has considered only afraction of the group presentations which may induce Cayley graph labelings withconditions at various distances.

In the preceding sections we constructed labelings based upon the choice of avery specific homomorphism of the group �. By considering labelings based on otherhomomorphisms, we may be able to show yet more groups yield Cayley graphs withoptimal spans.

For instance, the unbalanced presentation 〈s, t|ts4 =st3〉 permits an L(2,1)-labeling� with span �=6 produced by letting �(s)=2 and �(t)=3 and extending as in the proofof the main theorem. This optimal labeling is obtainable since these choices for �(s)and �(t) satisfy

4�(s)+�(t)≡�(s)+3�(t) mod 7.

We can analyze such 2-generator, 1-relator groups more generally: without loss ofgenerality we may write any single relator w=w′ as w(w′)−1 =1, so we assume ourgroup is given by the presentation 〈s, t|w=1〉. Let es =exps(w) and et =expt(w). Shouldit be the case that one of

2es +3et,3es +2et,2es +4et or 4es +2et

is congruent to 0 mod 7, then we may define a homomorphism � which yields anL(2,1)-labeling with span �=6. By exhausting all cases we checked that one of theseequivalences holds in 25 of the 49 possible choices for (exps(w), expt(w)) after reductionmodulo 7.

The preceding paragraph can be generalized to the following observation, whoseproof is nearly identical to the argument in Section 2:

Proposition 4.1. Let G be the Cayley graph corresponding to a 2-generator, 1-relatorpresentation. In 25

49 ≈0.5102 of the cases of (exps(w), expt(w)), G can be given anL(2,1)-labeling � with (optimal ) span �=6.

Indeed, the preceding result is a special case of the following much more generalobservation, whose proof is nearly identical to the argument in Section 2:

Proposition 4.2. Let G be the 2n-regular undirected graph underlying the Cayleygraph corresponding to the involution-free presentation 〈S|w〉 (for S={s1,s2, . . . ,sn}and w a word in the alphabet S±1), and define ei to be the exponent sum of si in w,modulo 2n+3. Let e= (e1,e2, . . . ,en)∈Zn

2n+3 be the presentation’s exponent vector, andsuppose there exists an assignment vector a= (a1,a2, . . . ,an) such that

(1) ai ∈{2,3, . . . ,2n,2n+1} for i=1, . . . ,n,(2) ai �=±aj mod 2n+3 if i �= j, and(3) e ·a≡0 mod 2n+3.

Then �(G;2,1)=2n+2.

Let us call any presentation satisfying the hypothesis of Proposition 4.2 semibal-anced. Numerical evidence suggests that semibalanced presentations are quite common.



For a given n, we let �(n)= (2n+3)n denote the number of potential exponent vectorse and �sb(n) denote the number of such vectors corresponding to semibalanced presen-tations as in Proposition 4.2.

Direct computation using Mathematica yields the following information for smallvalues of n:

n �(n) �sb(n)�sb(n)�(n)

1 5 1 0.200002 49 25 0.510203 729 603 0.827164 14641 14481 0.989075 371293 370993 0.99919

Thus the proportion of exponent vectors corresponding to semibalanced presentationsappears to converge very quickly to 1. These data suggest the following

Conjecture 4.3. The ratio of the number of Cayley graphs G corresponding ton-generator, 1-relator presentations with �(G;2,1)=2n+2 to the number of Cayleygraphs corresponding to any such presentation approaches 1 as n→∞.

Proving this conjecture will require, for a given n, estimating the number of exponentvectors orthogonal (mod 2n+3) to some assignment vector. This may be easier when2n+3 is prime, in which case the discrete Fourier analytic methods of Iosevich andSenger in [8] can apply. These authors have already indicated that the problem is likelya tractable one [9] and have begun collaboration with the present author in order tocomplete a proof of the conjecture.

To conclude we mention another generalization, this one of the graph theoreticconditions imposed on the labelings. Griggs and Yeh [5] consider conditions atp distinct distances, posing the problem of finding the span �(G;k1, . . . ,kp) of anoptimal L(k1, . . . ,kp)-labeling. It is clear that though the computations will becomemessier, the approach undertaken in this article can be generalized to produce labelings� that give bounds for these more general spans �. Although there is no theoreticalobstacle to this generalization, describing the desired group homomorphisms willlikely become much more difficult except in very specific cases.

ACKNOWLEDGMENTS

The author was supported by NSF REU Grant DMS-0647804, “Groups, Graphs, andGeometry,” during the writing of this article.

REFERENCES

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[3] J. P. Georges and D. W. Mauro, Generalized vertex labelings with a conditionat distance two, Congr Numer 109 (1995), 141–159.

[4] J. R. Griggs and X. T. Jin, Real number channel assignment for lattices,SIAM J Discrete Math 22(3) (2008), 996–1021.

[5] J. R. Griggs and R. K. Yeh, Labelling graphs with a condition at distance 2,SIAM J Discrete Math 5(4) (1992), 586–595.

[6] W. K. Hale, Frequency assignment: theory and applications, Proc IEEE 68(1980), 1497–1514.

[7] J. van den Heuvel, R. A. Leese, and M. A. Shepherd, Graph labeling andradio channel assignment, J Graph Theory 29 (1998), 263–283.

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Channel assignment on Cayley graphs