Distance-constrained labellings of Cartesian products of graphs
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 and nonnegative integers , determine the smallest positive integer with the property that each vertex of can be assigned a label from such that for every pair of vertices at distance receive labels which differ by at least . The -number of , denoted by , is defined as the smallest positive integer with this property. This parameter and several variants of it, especially for small , have received much attention in the past more than three decades. In particular, a number of results on the -number have been produced by many researchers, especially in the case when , 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 for any graph with maximum degree . 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 -number has also received considerable attention (see, for example, [2], [5], [16], [17], [28]), but for very little is known about the -number. In general, it is difficult to determine the exact value of for a general graph. For example, for , we have , and so determining is equivalent to computing the chromatic number . Answering a question posed in [17], it was proved in [5] that the problem of determining the -number is NP-complete even for trees.
In this paper we study the -number and three variants of it (see Section 1.2 for their definitions) for any graph which is the Cartesian product of non-trivial graphs, where and is an -tuple with . We prove that under a certain condition these four invariants for all attain a common lower bound, and in particular the chromatic number of the th power of is equal to this lower bound plus . We obtain further a sandwich theorem which says that under the same condition the same result holds for every subgraph of that contains a certain subgraph of as a subgraph. As corollaries we obtain that these results are true for Hamming graphs such that for some with . 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 and three variants of it can be computed exactly.
All graphs considered in the paper are finite, undirected and simple. Let be a graph with vertex set and edge set . As usual denote by the chromatic number of and call the order of . Denote by the distance in between vertices of . For an integer , the th power of is the graph with vertex set in which are adjacent if and only if . We write to denote that is a subgraph of .
The Cartesian product of given graphs , denoted by , is the graph with vertex set in which two vertices , are adjacent if and only if there is exactly one such that and for all .
Given integers , the Hamming graph is the Cartesian product where, for an integer , denotes the complete graph with order . Since the Cartesian product is commutative, without loss of generality we may assume that . In the case when , we write in place of . In particular, is the -dimensional hypercube .
Let be nonnegative integers. An -labelling of is a mapping from to the set of nonnegative integers such that, for and any pair of vertices with , The integer is the label of under and the span of , denoted by , is the difference between the largest and smallest labels assigned to the vertices of by . Without loss of generality we may always assume that the smallest label used is , so that The -number of is defined [7], [10] as where the minimum is taken over all -labellings of . Equivalently, as stated in the beginning of this paper, is the smallest positive integer such that an -labelling of with span exists.
The above notion of distance labelling originated from the frequency assignment problem [11] for which the value measures the minimum bandwidth required by a radio communication network modelled by under the constraints (1). It is readily seen that where is an -tuple. Thus, from a pure graph-theoretical point of view, the -labelling problem can be considered as a generalization of the classical vertex-colouring problem.
An -labelling of is said to be no-hole (see, for example, [3], [4], [21], [23], [24]) if is a set of consecutive integers. Define to be the minimum span among all no-hole -labellings of , and if no such a labelling exists. As an example, we see that for all and .
The -labelling problem and its no-hole version are a linear model in the sense that the -metric is used to measure the span between two channels. The cyclic version of the -labelling problem was studied in [13] with a focus on small . A mapping is called a -labelling of with span if, for and any with , where is the -cyclic distance between and . A -labelling of with span exists for sufficiently large . The -number of , denoted by , is defined to be the minimum integer such that admits a -labelling with span . Note that thus defined agrees with defined in [3] but is one less than used in [13] and used in [19]. As observed in [6], [13], this cyclic version allows the assignment of a set of channels to each transmitter when is viewed as a radio network with one transmitter placed at each vertex.
A -labelling of with span is no-hole if is a set of consecutive integers . Define to be the minimum such that admits a no-hole -labelling of span , and if no such a labelling exists.
It can be verified that if then the four invariants above are all monotonically increasing; that is, for whenever is a subgraph of .
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 for some between and , then In [9, Theorem 3.1] it was shown that, if is a prime and either and , or and , then The -number of was determined in [9] and results on can be found in [8], [20].
In [26] a group-theoretic approach to -labelling Cayley graphs of Abelian groups was introduced. As an application it was proved [26] among other things that for any if , divides and is no less than , and every prime factor of is no less than , generalizing (3) to a wide extent. In [26] it was also proved (as a corollary of a more general result) that , which yields when , where and . In the special case when , the upper bound (4) gives exactly (2) (see [26, p. 990] for justification). In [28] lower and upper bounds on were obtained using a group-theoretic approach, which recover the main result in [15] in the special case when . The problem of determining , or equivalently the chromatic number of powers of , 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 for as contains as a subgraph and has order and diameter two. The following question was asked in [26, Question 6.1] (see also [4, Section 5]): Given integers and with , for which integers with do we have ? A partial answer to this question was given in [4, Theorem 1.3], where it was proved that, for , if is sufficiently large, namely , where is the largest subscript such that , then This result inspired us to explore when a similar phenomenon occurs for , , and 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 and be non-trivial graphs with orders and , respectively, and let , where . Let be an integer with . Suppose that and contains a subgraph with order and diameter at most . Then for any integer with we have where is an -tuple. Moreover, there is a labelling of that is optimal for , , and simultaneously. In particular, we have and the same labelling gives rise to an optimal colouring of .
Since by our assumption contains a subgraph with order and diameter at most , it is easy to see that is a lower bound for each of the four invariants in (5). Theorem 1.1 asserts that actually these four invariants for 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 to . Recall that we write when is a subgraph of a graph .
Theorem 1.2 Sandwich Theorem Under the conditions of Theorem 1.1, for every graph with and any integer with , we have where is an -tuple. Moreover, there is a labelling of that is optimal for , , and simultaneously. In particular, we have and the same labelling gives rise to an optimal colouring of .
Setting and in Theorem 1.1, we have . Since has subgraph with order and diameter , Theorem 1.1 implies immediately the following result for Hamming graphs.
Corollary 1.3 Let be integers no less than , and let be an integer with . Let . Suppose that Then for any integer with we have where is an -tuple. Moreover, there is a labelling of that is optimal for , , and simultaneously. In particular, we have and the same labelling gives rise to an optimal colouring of .
Similarly, Theorem 1.2 implies the following result, in which the claimed optimal labelling is the restriction of the above mentioned optimal labelling of to .
Corollary 1.4 Sandwich Theorem for Hamming Graphs Under the conditions of Corollary 1.3, for every graph such that and any integer with , we have where is an -tuple. Moreover, there is a labelling of that is optimal for , , and simultaneously. In particular, we have and the same labelling gives rise to an optimal colouring of .
Corollary 1.3 implies the following result for Hamming graphs and the -dimensional hypercube .
Corollary 1.5 Let and be integers such that , and Then for any integer with we have where is an -tuple. In particular, if , then for ,
Note that (9) requires . This is so because for the inequalities in (8) cannot be true unless . Similarly, for , (8) requires . In the general case when , (8) says that is between and .
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 be a graph and let be nonnegative integers. Then
Corollary 2.2 Let be a graph and let be nonnegative integers. If = , then and any optimal no-hole
A lemma
Lemma 3.1 Let and be non-trivial graphs with orders and , respectively, and let , where . If then for any integer with , we have where is an -tuple.
Proof Since and are non-trivial graphs, their orders and are no less than . Since the Cartesian product is commutative, without loss of generality we may assume that , so (13) becomes .
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 be integers no less than such that and , where . Let be an integer between and . Then . Take an integer such that . Then and . Let , and let
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 .
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