Elsevier

Discrete Applied Mathematics

Volume 304, 15 December 2021, Pages 375-383
Discrete Applied Mathematics

Distance-constrained labellings of Cartesian products of graphs

https://doi.org/10.1016/j.dam.2021.08.012Get rights and content

Abstract

An L(h1,h2,,hl)-labelling of a graph G is a mapping ϕ:V(G){0,1,2,} such that for 1il and each pair of vertices u,v of G at distance i, we have |ϕ(u)ϕ(v)|hi. The span of ϕ is the difference between the largest and smallest labels assigned to the vertices of G by ϕ, and λh1,h2,,hl(G) is defined as the minimum span over all L(h1,h2,,hl)-labellings of G.

In this paper we study λh,1,,1 for Cartesian products of graphs, where (h,1,,1) is an l-tuple with l3. We prove that, under certain natural conditions, the value of this and three related invariants on a graph H which is the Cartesian product of l graphs attain a common lower bound. In particular, the chromatic number of the lth power of H equals this lower bound plus one. We further obtain a sandwich theorem which extends the result to a family of subgraphs of H which contain a certain subgraph of H. All these results apply in particular to the class of Hamming graphs: if q1qd2 and 3ld then the Hamming graph H=Hq1,q2,,qd satisfies λql,1,,1(H)=q1q2ql1 whenever q1q2ql1>3(ql1+1)qlqd. In particular, this settles a case of the open problem on the chromatic number of powers of the hypercubes.

Introduction

Motivated by the frequency assignment problem [6], [11] for communication networks, various optimal labelling problems for graphs involving distance conditions have been studied extensively since the 1980s. Among them is the following well-known distance labelling problem: Given a graph G and nonnegative integers h1,h2,,hl, determine the smallest positive integer k with the property that each vertex of G can be assigned a label from {0,1,,k} such that for 1il every pair of vertices at distance i receive labels which differ by at least hi. The λh1,h2,,hl-number of G, denoted by λh1,h2,,hl(G), is defined as the smallest positive integer k with this property. This parameter and several variants of it, especially for small l, have received much attention in the past more than three decades. In particular, a number of results on the λh1,h2-number have been produced by many researchers, especially in the case when (h1,h2)=(2,1), as one can find in the survey paper [1]. Much work in this case was motivated by a conjecture of Griggs and Yeh [10] which asserts that λ2,1(G)Δ2 for any graph G with maximum degree Δ. As far as we know, this conjecture is still open in its general form, though it has been confirmed in many special cases (see, for example, [8], [10], [12], [14], [22]). In recent years, the λh1,h2,h3-number has also received considerable attention (see, for example, [2], [5], [16], [17], [28]), but for l>3 very little is known about the λh1,h2,,hl-number. In general, it is difficult to determine the exact value of λh1,h2,,hl for a general graph. For example, for l=1, we have λh1(G)=h1(χ(G)1), and so determining λh1 is equivalent to computing the chromatic number χ. Answering a question posed in [17], it was proved in [5] that the problem of determining the λh1,1,1-number is NP-complete even for trees.

In this paper we study the λh,1,,1-number and three variants of it (see Section 1.2 for their definitions) for any graph H which is the Cartesian product of l non-trivial graphs, where l3 and (h,1,,1) is an l-tuple with h1. We prove that under a certain condition these four invariants for H all attain a common lower bound, and in particular the chromatic number of the lth power of H is equal to this lower bound plus 1. We obtain further a sandwich theorem which says that under the same condition the same result holds for every subgraph of H that contains a certain subgraph of H as a subgraph. As corollaries we obtain that these results are true for Hamming graphs Hq1,q2,,qd such that q1q2ql1>3(min{q1,,ql1}+1)qlqd for some l with 3l<d. We will give the precise statements of our results in Theorem 1.1, Theorem 1.2 and Corollary 1.3, Corollary 1.4, Corollary 1.5 after introducing relevant definitions and giving a brief review of related results in Sections 1.2 Distance-constrained labelling problems, 1.3 Distance-constrained labellings of Hamming graphs, respectively. Our results give infinite families of graphs for which the values of λh,1,,1 and three variants of it can be computed exactly.

All graphs considered in the paper are finite, undirected and simple. Let G be a graph with vertex set V(G) and edge set E(G). As usual denote by χ(G) the chromatic number of G and call |V(G)| the order of G. Denote by distG(u,v) the distance in G between vertices u,v of G. For an integer l1, the lth power Gl of G is the graph with vertex set V(G) in which u,vV(G) are adjacent if and only if 1distG(u,v)l. We write KG to denote that K is a subgraph of G.

The Cartesian product of given graphs G1,G2,,Gd, denoted by G1G2Gd, is the graph with vertex set V(G1)×V(G2)××V(Gd) in which two vertices (u1,u2,,ud), (v1,v2,,vd) are adjacent if and only if there is exactly one i{1,2,,d} such that uiviE(Gi) and uj=vj for all j{1,2,,d}{i}.

Given integers q1,q2,,qd2, the Hamming graph Hq1,q2,,qd is the Cartesian product Kq1Kq2Kqd where, for an integer q1, Kq denotes the complete graph with order q. Since the Cartesian product is commutative, without loss of generality we may assume that q1q2qd. In the case when q1=q2==qd=q, we write H(d,q) in place of Hq1,q2,,qd. In particular, H(d,2) is the d-dimensional hypercube Qd.

Let h1,h2,,hl be nonnegative integers. An L(h1,h2,,hl)-labelling of G is a mapping ϕ from V(G) to the set of nonnegative integers such that, for i=1,2,,l and any pair of vertices u,vV(G) with distG(u,v)=i, |ϕ(u)ϕ(v)|hi.The integer ϕ(u) is the label of u under ϕ and the span of ϕ, denoted by sp(G;ϕ), is the difference between the largest and smallest labels assigned to the vertices of G by ϕ. Without loss of generality we may always assume that the smallest label used is 0, so that sp(G;ϕ)=maxvV(G)ϕ(v).The λh1,h2,,hl-number of G is defined [7], [10] as λh1,h2,,hl(G)=minϕsp(G;ϕ),where the minimum is taken over all L(h1,h2,,hl)-labellings of G. Equivalently, as stated in the beginning of this paper, λh1,h2,,hl(G) is the smallest positive integer k such that an L(h1,h2,,hl)-labelling of G with span k exists.

The above notion of distance labelling originated from the frequency assignment problem [11] for which the value λh1,h2,,hl(G) measures the minimum bandwidth required by a radio communication network modelled by G under the constraints (1). It is readily seen that χ(Gl)=λ1,1,,1(G)+1,where (1,1,,1) is an l-tuple. Thus, from a pure graph-theoretical point of view, the L(h1,h2,,hl)-labelling problem can be considered as a generalization of the classical vertex-colouring problem.

An L(h1,h2,,hl)-labelling ϕ of G is said to be no-hole (see, for example, [3], [4], [21], [23], [24]) if {ϕ(v):vV(G)} is a set of consecutive integers. Define λ¯h1,h2,,hl(G) to be the minimum span among all no-hole L(h1,h2,,hl)-labellings of G, and if no such a labelling exists. As an example, we see that λ¯h1,h2,h3(Kq)= for all h12 and q2.

The L(h1,h2,,hl)-labelling problem and its no-hole version are a linear model in the sense that the L1-metric is used to measure the span between two channels. The cyclic version of the L(h1,h2,,hl)-labelling problem was studied in [13] with a focus on small l. A mapping ϕ:V(G){0,1,2,,k1} is called a C(h1,h2,,hl)-labelling of G with span k if, for i=1,2,,l and any u,vV(G) with distG(u,v)=i, |ϕ(u)ϕ(v)|khi,where |xy|k=min{|xy|,k|xy|}is the k-cyclic distance between x and y. A C(h1,h2,,hl)-labelling of G with span k exists for sufficiently large k. The σh1,h2,,hl-number of G, denoted by σh1,h2,,hl(G), is defined to be the minimum integer k1 such that G admits a C(h1,h2,,hl)-labelling with span k. Note that σh1,h2,,hl(G) thus defined agrees with σ(G;h1,h2,,hl) defined in [3] but is one less than σ(G;h1,h2,,hl) used in [13] and ch1,h2,,hl(G) used in [19]. As observed in [6], [13], this cyclic version allows the assignment of a set of channels ϕ(u),ϕ(u)+k,ϕ(u)+2k, to each transmitter u when G is viewed as a radio network with one transmitter placed at each vertex.

A C(h1,h2,,hl)-labelling ϕ of G with span k is no-hole if {ϕ(v):vV(G)} is a set of consecutive integers modk. Define σ¯h1,h2,,hl(G) to be the minimum k1 such that G admits a no-hole C(h1,h2,,hl)-labelling of span k, and if no such a labelling exists.

It can be verified that if h1h2hl then the four invariants above are all monotonically increasing; that is, η(H)η(G) for η=λh1,h2,,hl,λ¯h1,h2,,hl,σh1,h2,,hl,σ¯h1,h2,,hl whenever H is a subgraph of G.

This paper was motivated by distance-constrained labellings of Hamming graphs and hypercubes. As such let us mention several known results on this class of graphs. More results can be found in the short survey [27].

In [25, Theorem 3.7] it was proved that, if 2n1d2nt for some t between 1 and n+1, then λ2,1(Qd)2n+2nt+12.In [9, Theorem 3.1] it was shown that, if p is a prime and either dp and r2, or d<p and r=1, then λ2,1(H(d,pr))=p2r1.The λj,k-number of Hq1,q2 was determined in [9] and results on Hq1,q2,q3 can be found in [8], [20].

In [26] a group-theoretic approach to L(j,k)-labelling Cayley graphs of Abelian groups was introduced. As an application it was proved [26] among other things that λj,k(Hq1,q2,,qd)=(q1q21)kfor any 2kjk1 if q1>d2, q2 divides q1 and is no less than q3,,qd, and every prime factor of q1 is no less than d, generalizing (3) to a wide extent. In [26] it was also proved (as a corollary of a more general result) that λj,k(Qd)2nmax{k,j/2}+2ntmin{jk,j/2}j, which yields λj,k(Qd)2nk+2nt(jk)jwhen 2kj, where n=1+log2d and t=min{2nd1,n}. In the special case when (j,k)=(2,1), the upper bound (4) gives exactly (2) (see [26, p. 990] for justification). In [28] lower and upper bounds on λh1,h2,h3(Qd) were obtained using a group-theoretic approach, which recover the main result in [15] in the special case when (h1,h2,h3)=(1,1,1). The problem of determining λ1,,1(Qd), or equivalently the chromatic number of powers of Qd, has a long history but is still wide open. See [18], [28] for some background information and related results. One of the contributions of the present paper settles this problem for a range of dimensions (see Corollary 1.5).

Note that λj,k(Hq1,q2,,qd)(q1q21)k for jk as Hq1,q2,,qd contains Hq1,q2 as a subgraph and Hq1,q2 has order q1q2 and diameter two. The following question was asked in [26, Question 6.1] (see also [4, Section 5]): Given integers j and k with 2kjk1, for which integers q1q2qd with j/kq1q2i=1dqi+d do we have λj,k(Hq1,q2,,qd)=(q1q21)k? A partial answer to this question was given in [4, Theorem 1.3], where it was proved that, for (j,k)=(2,1),(1,1), if q1 is sufficiently large, namely q1d+n1+i=2d(i2)(qi1), where n is the largest subscript such that q2=qn, then λj,k(Hq1,q2,,qd)=λ¯j,k(Hq1,q2,,qd)=σ¯j,k(Hq1,q2,,qd)=σj,k(Hq1,q2,,qd)=q1q21.This result inspired us to explore when a similar phenomenon occurs for λh,1,,1, λ¯h,1,,1, σ¯h,1,,1 and σh,1,,1 for Cartesian products of graphs. As will be seen in the next subsection, our main results in the present paper provide sufficient conditions for this to happen.

The first main result in this paper is as follows.

Theorem 1.1

Let G1,,Gl1 and G be non-trivial graphs with orders q1,,ql1 and q, respectively, and let H=G1Gl1G, where l3. Let ql be an integer with 1qlq. Suppose that q1q2ql1>3(min{q1,,ql1}+1)qand H contains a subgraph K with order q1q2ql and diameter at most l. Then for any integer h with 1hql we have λh,1,,1(H)=λ¯h,1,,1(H)=σ¯h,1,,1(H)=σh,1,,1(H)=q1q2ql1,where (h,1,,1) is an l-tuple. Moreover, there is a labelling of H that is optimal for λh,1,,1, λ¯h,1,,1, σ¯h,1,,1 and σh,1,,1 simultaneously. In particular, we have χ(Hl)=q1q2qland the same labelling gives rise to an optimal colouring of Hl.

Since by our assumption H contains a subgraph with order q1q2ql and diameter at most l, it is easy to see that q1q2ql1 is a lower bound for each of the four invariants in (5). Theorem 1.1 asserts that actually these four invariants for H all achieve this trivial lower bound. In Section 4, we will give a general construction to show that there are many graphs other than Hamming graphs which satisfy the conditions of Theorem 1.1.

Using Theorem 1.1, we obtain the following sandwich result, which is our second main result in the paper. The claimed optimal labelling in this sandwich theorem is the restriction of the above mentioned optimal labelling of H to V(X). Recall that we write KG when K is a subgraph of a graph G.

Theorem 1.2 Sandwich Theorem

Under the conditions of Theorem 1.1, for every graph X with KXH and any integer h with 1hql, we have λh,1,,1(X)=λ¯h,1,,1(X)=σ¯h,1,,1(X)=σh,1,,1(X)=q1q2ql1,where (h,1,,1) is an l-tuple. Moreover, there is a labelling of X that is optimal for λh,1,,1, λ¯h,1,,1, σ¯h,1,,1 and σh,1,,1 simultaneously. In particular, we have χ(Xl)=q1q2qland the same labelling gives rise to an optimal colouring of Xl.

Setting G1=Kq1,,Gl1=Kql1 and G=KqlKqd in Theorem 1.1, we have H=G1Gl1G=Hq1,q2,,qd. Since Hq1,q2,,qd has subgraph Hq1,q2,,ql with order q1q2ql and diameter l, Theorem 1.1 implies immediately the following result for Hamming graphs.

Corollary 1.3

Let q1q2qd be integers no less than 2, and let l be an integer with 3l<d. Let H=Hq1,q2,,qd. Suppose that q1q2ql1>3(ql1+1)qlqd.Then for any integer h with 1hql we have λh,1,,1(H)=λ¯h,1,,1(H)=σ¯h,1,,1(H)=σh,1,,1(H)=q1q2ql1,where (h,1,,1) is an l-tuple. Moreover, there is a labelling of H that is optimal for λh,1,,1, λ¯h,1,,1, σ¯h,1,,1 and σh,1,,1 simultaneously. In particular, we have χ(Hl)=q1q2qland the same labelling gives rise to an optimal colouring of Hl.

Similarly, Theorem 1.2 implies the following result, in which the claimed optimal labelling is the restriction of the above mentioned optimal labelling of Hq1,q2,,qd to V(X).

Corollary 1.4 Sandwich Theorem for Hamming Graphs

Under the conditions of Corollary 1.3, for every graph X such that Hq1,q2,,qlXHq1,q2,,qd and any integer h with 1hql, we have λh,1,,1(X)=λ¯h,1,,1(X)=σ¯h,1,,1(X)=σh,1,,1(X)=q1q2ql1,where (h,1,,1) is an l-tuple. Moreover, there is a labelling of X that is optimal for λh,1,,1, λ¯h,1,,1, σ¯h,1,,1 and σh,1,,1 simultaneously. In particular, we have χ(Xl)=q1q2qland the same labelling gives rise to an optimal colouring of Xl.

Corollary 1.3 implies the following result for Hamming graphs H(d,q) and the d-dimensional hypercube Qd=H(d,2).

Corollary 1.5

Let d,q and l be integers such that d6, q2 and (d+4+max{4q,0})/2l<d.Then for any integer h with 1hq we have λh,1,,1(H(d,q))=ql1,where (h,1,,1) is an l-tuple. In particular, if (d+6)/2l<d, then for h=1,2, λh,1,,1(Qd)=2l1.

Note that (9) requires d8. This is so because for q=2 the inequalities in (8) cannot be true unless d8. Similarly, for q=3, (8) requires d7. In the general case when q4, (8) says that l is between (d+4)/2 and d1.

Theorem 1.1, Theorem 1.2 will be proved in Section 3 after a short preparation in the next section. The paper concludes in Section 4 with a construction illustrating the wide applicability of Theorem 1.1, Theorem 1.2, some final remarks assessing the strength of the sufficient conditions in Theorem 1.1 and two open problems.

Section snippets

Preliminaries

The following inequalities follow immediately from related definitions (see [13] for the inequalities in (10)).

Lemma 2.1

Let G be a graph and let h1h2hl be nonnegative integers. Then

λh1,h2,,hl(G)σh1,h2,,hl(G)λh1,h2,,hl(G)+h11 λh1,h2,,hl(G)λ¯h1,h2,,hl(G)σ¯h1,h2,,hl(G) σh1,h2,,hl(G)σ¯h1,h2,,hl(G).

Corollary 2.2

Let G be a graph and let h1h2hl be nonnegative integers. If λh1,h2,,hl(G) = σ¯h1,h2,,hl(G), then λh1,h2,,hl(G)=λ¯h1,h2,,hl(G)=σ¯h1,h2,,hl(G)=σh1,h2,,hl(G)and any optimal no-hole C(h1,h

A lemma

Lemma 3.1

Let G1,,Gl1 and G be non-trivial graphs with orders q1,,ql1 and q, respectively, and let H=G1Gl1G, where l3. If q1q2ql1>3(min{q1,,ql1}+1)q,then for any integer ql with 1qlq, we have σ¯ql,1,,1(H)q1q2ql1,where (ql,1,,1) is an l-tuple.

Proof

Since G1,,Gl1 and G are non-trivial graphs, their orders q1,,ql1 and q are no less than 2. Since the Cartesian product is commutative, without loss of generality we may assume that ql1=min{q1,,ql1}, so (13) becomes q1q2ql1>3(ql1+1)q.

Concluding remarks

It is not difficult to construct many graphs other than Hamming graphs which satisfy the conditions of Theorem 1.1, and we give a simple construction here. Let q1,,ql1 be integers no less than 2 such that ql1=min{q1,,ql1} and 3ql1(ql1+1)<q1, where l3. Let ql be an integer between 1 and ql1. Then q2ql1ql<(q1q2ql2ql)/(3(ql1+1)). Take an integer q such that q2ql1qlq<(q1q2ql2ql)/(3(ql1+1)). Then 1qlql1<q and 3(ql1+1)q<q1q2ql2qlq1q2ql2ql1. Let G1=Kq1, and let G2,,Gl1

Acknowledgements

We would like to thank the anonymous referees for their helpful comments. Anna Lladó and Oriol Serra acknowledge financial support from the Spanish Agencia Estatal de Investigación under project MTM2017-82166-P. Zhou was supported by the Research Grant Support Scheme of The University of Melbourne, Australia .

References (28)

  • ZhouS.

    A distance-labelling problem for hypercubes

    Discrete Appl. Math.

    (2008)
  • CalamonleriT.

    The L(hk)-labelling problem: An updated survey and annotated bibliography

    Comput. J.

    (2011)
  • CalamonleriT. et al.

    L(h,1,1)-labelling of outplanar graphs

    Math. Methods Oper. Res.

    (2009)
  • GamstA.

    Homogenous distribution of frequencies in a regular hexagonal cell system

    IEEE Trans. Veh. Technol.

    (1982)
  • Cited by (1)

    View full text