Decomposing complete equipartite multigraphs into cycles of variable lengths: the amalgamation-detachment approach

09/20/2019 ∙ by Amin Bahmanian, et al. ∙ uOttawa 0

Using the technique of amalgamation-detachment, we show that the complete equipartite multigraph λ K_n× m can be decomposed into cycles of lengths c_1m,...,c_km (plus a 1-factor if the degree is odd) whenever there exists a decomposition of λ m K_n into cycles of lengths c_1, ...,c_k (plus a 1-factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph λ K_n× m.

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1 Introduction

A decomposition of a graph is a collection of subgraphs of whose edge sets partition the edge set of . A graph with a cycle decomposition has no vertices of odd degree, however, a graph in which every vertex has odd degree may admit a decomposition into cycles and a 1-factor. By a -cycle decomposition of a graph we shall mean a decomposition of into cycles of lengths , respectively, if every vertex in has even degree, and a decomposition of into cycles of lengths plus a 1-factor if every vertex in has odd degree.

In this paper, we are concerned with decompositions of complete multigraphs and complete equipartite multigraphs into cycles of variable lengths (plus a 1-factor if the vertex degrees are odd). In particular, we show how cycle decompositions of complete multigraphs can be used to obtain cycle decompositions of complete equipartite multigraphs.

Bryant, Horsley, and Pettersson [10] recently proved the following result, which had been conjectured over thirty years ago by Alspach [1].

Theorem 1.1

[10] The complete graph admits a -cycle decomposition if and only if and .

Can this result be generalized to complete multigraphs? It is easy to see that if there exists a -cycle decomposition of the complete multigraph , then the following necessary conditions hold:

(B1)

for all and

(B2)

.

In the case of cycles of length 2, two more necessary conditions are required (see [9, 8] and Lemma 3.1):

(B3)

If is odd, then ; and

(B4)

if is even, then .

Bryant, Horsley, Maenhaut, and Smith [9] showed that Conditions (B1)–(B3) are also sufficient in the following cases.

Theorem 1.2

[9] Let , , , and be positive integers satisfying Conditions (B1)–(B3), with . In addition, assume that

  1. ; or

  2. ; or

  3. .

Then the complete multigraph admits a -cycle decomposition.

It appears that is has now been proved [8] that Conditions (B1)–(B4) are in fact sufficient in all cases.

In this paper, we are concerned with the following generalization of Alspach’s Conjecture.

Problem 1.3

Determine the necessary and sufficient conditions on parameters , , , and for the complete equipartite multigraph (with parts of cardinality ) to admit a -cycle decomposition.

Observe that Conditions (C1)–(C4) below are necessary for the existence of a -cycle decomposition of . While Conditions (C1)–(C3) are easy to see, Condition (C4) will be proved in Lemma 3.1 in more generality.

(C1)

for all ;

(C1)

if , then are all even;

(C2)

;

(C3)

if is odd, then ; and

(C4)

if is even, then .

Many partial solutions to Problem 1.3 are known for complete equipartite graphs of even degree and uniform cycle lengths, that is, for . Necessary and sufficient conditions have been determined for [15]; [7]; [4]; prime [22]; twice a prime [26], three times a prime [25], and prime square [24]; and for small relative to the number of parts [27, 28, 18]. Problem 1.3 has also been completely solved for graphs of even degree and uniform cycle lengths when the number of parts is small; namely for [29]; [11], [6], and [5]. For complete equipartite multigraphs of even degree and uniform cycle lengths, Problem 1.3 has been solved for [7] and prime [23].

Variable (but very specific) cycle lengths in the bipartite graph were considered in [2, 14, 12, 18, 21], and in the bipartite multigraph , in [13]. The most comprehensive result to date, contained in [18], solves Problem 1.3 for cycle lengths satisfying .

For complete equipartite multigraphs with , the only known result for variable cycle lengths gives necessary and sufficient conditions when , is even, and for all [19]. No results are known for the case , , and variable cycle lengths.


The main goal of this paper is to offer a partial solution to Problem 1.3 in the following form.

Theorem 1.4

Let , , , and be positive integers such that there exists a -cycle decomposition of . Then the complete equipartite multigraph admits a -cycle decomposition.

The following corollary is immediate.

Corollary 1.5

Let , , , and be positive integers, and let . Assume that , , , and satisfy the conditions of Theorem 1.2. Then the complete equipartite multigraph admits a -cycle decomposition.

This paper is organized as follows. In Section 2 we give the necessary definitions, terminology, and technical tools that will be used in Section 3 to prove our main result, Theorem 1.4. The techniques used in its proof are taken a step further to construct some other, related cycle decompositions of complete equipartite multigraphs in Section 4. Finally, in Section 5, we give all possible cycle decompositions of the complete multigraphs with at most four vertices, and as a corollary using Theorem 1.4, all possible decompositions of the complete equipartite multigraphs with at most four parts into cycles of lengths divisible by .

2 Preliminaries

All graphs in this paper are assumed to be finite, loopless, and undirected, often with multiple edges. As usual, the symbol denotes the complete graph with vertices, and denotes the complete -partite graph with all parts of cardinality . For any simple graph and positive integer , the symbol denotes the multigraph with the multiplicity of every edge equal to and with the underlying simple graph isomorphic to .

Let be a graph, and and two distinct vertices of . Then , , and will denote the degree of , the set of neighbours of , and the number of edges between and , respectively, in . Similarly, if , then denotes the number of edges of incident with and a vertex in . If , then denotes the graph obtained from by deleting all vertices in , as well as all edges incident with a vertex in . If , then we write simply instead of .

Two concepts will be used in the proof of Theorem 1.4 as the main tools. The first is the amalgamation-detachment technique, first developed in [16, 17], and more recently surveyed in [3]. Informally speaking, an -detachment of a graph is any graph obtained by splitting a vertex of into one or more vertices, and dividing the edges incident with among the resulting (sub)vertices. In particular, in an -detachment of in which we split vertex into vertices and , each edge of the form in will give rise to an edge of the form either or in .

The following lemma will be crucial in the induction step of the proof of Theorem 3.2.

Lemma 2.1

Let be a connected graph, and be an -detachment of obtained by splitting a vertex into two vertices and . Then is connected if and only if for some connected component of .

Proof. Let and be as in the statement of the lemma. Note that the connected components of are precisely the connected components of . If is connected, then some component of must contain neighbours of both and , and conversely. In other words, is connected if and only if and for some connected component of . Since , the statement of the lemma then follows immediately. a

The second main tool in the proof of Theorem 1.4 is the concept of edge colouring, in particular, de Werra’s Theorem 2.2 below. A -edge-colouring of a graph is a mapping , where is the set of colours. For any , the symbol will denote the spanning subgraph of whose edge set is the set of all edges of colour ; we call such a spanning subgraph a colour class of with respect to the edge colouring . Observe that a colour class may have (many) isolated vertices.

A -edge-colouring of a graph is called equitable if for all and ; that is, if every vertex is incident with “almost the same” number of edges of each colour.

The following extremely useful result by de Werra [30] guarantees existence of an equitable -edge-colouring in any bipartite graph. For completeness, and since publication [30] is not available to us, we present a proof.

Theorem 2.2

[30] Let be a bipartite graph and a positive integer. Then admits an equitable -edge-colouring.

Proof. The assertion clearly holds for , hence we may assume .

For a graph and a fixed -edge-colouring of , let and for all and . Define a parameter as follows:

Claim: if and only if is an equitable -edge-colouring of .

Proof of the claim: Assume is an equitable -edge-colouring of . Then, for all and , we have , and it easily follows that .

Conversely, let be a -edge-colouring of with . First observe that, for any real number , the quantity gives the sum of distances of from 0 and 1, and hence . Thus implies that for all and . Since is a non-negative integer, it follows that , in other words, is an equitable -edge-colouring of .

Now suppose is a bipartite graph that admits no equitable -edge-colouring. Let be a -edge-colouring of that minimizes . Since is not equitable, there exist a vertex and colours such that . Let be a maximal trail in with initial vertex , first edge in , and edges alternately in and . (Recall that a trail is an alternating sequence of vertices and edges such that each edge has endpoints and , and no edge in the sequence is repeated.) For any internal vertex of , the trail enters with an edge of colour and exits with an edge of colour , or vice-versa. Since is bipartite, , and is maximal, the trail cannot be closed. Thus its terminal vertex, call it , is distinct from .

We construct a new -edge-colouring of by swapping colours and of along the trail . With respect to this new colouring , let and for all and .

We now show that

(1)

for all , with strict inequality when . This is obvious (with equality) for vertices since for all . It is also clear that for ,

(2)

holds (with equality) for all .

We now verify that strict inequality holds in (2) for and . Since , we have . Note that as a consequence, if , and otherwise. In any case,

Next, we verify Inequality (2) for and . Since is a maximal trail alternating colours and , for the terminal vertex of , we must have either or . Assume . Then the last edge of the trail must be of colour , and will be swapped to colour in . Hence if , and otherwise. Furthermore, if , and otherwise. In any case,

A similar argument shows that Inequality (2) holds for and when .

Finally, we’ll show that

(3)

for and any .

Take any colour . Since and , we have

Therefore, since , we can see that

Furthermore, since and , we also have and . Thus

unless both and . Now if and only if either or both and ; that is, if and only if either or both and . Similarly, if and only if either or both and . The only possibility then is . In this case, we must have , , and , and hence

Hence Inequality (3) holds for . Similarly, reversing the roles of colours and if necessary, we can show that it holds for .

We have thus shown that (1) holds for all , with strict inequality when . We conclude that is a -edge-colouring of with , a contradiction. Hence must possess an equitable -edge-colouring. a

3 Proof of the main result

Throughout the rest of this paper, unless otherwise specified, , , , , , and will denote positive integers,

We first give a simple lemma that implies the necessary condition (C4) for existence of a -cycle decomposition of .

Lemma 3.1

[8] Let be a multigraph in which each edge has even multiplicity, and assume that admits a -cycle decomposition. Then

Proof. Suppose that the result does not hold, and let be a smallest counterexample. That is, is a multigraph with the smallest number of edges such that every edge of has even multiplicity, and admits a -cycle decomposition with . Let and , and let be a cycle in of maximum length . For each , choose an edge parallel to , , and let . Since , we have . Hence there exists a cycle in , , that contains at most one edge of . If contains no edges of , then is a -cycle parallel to , , and each of the remaining cycles of contains at least 2 edges of . It follows that , contradicting .

Hence must contain exactly one edge of — call it — and the edges in form a path of length . Let be the path in parallel to . Obtain a graph from by deleting the edges of and , and a cycle decomposition of by deleting from and replacing with the cycle . Observe that has edges, and each edge has even multiplicity. Moreover, is indeed a cycle decomposition of ; it contains cycles, and maximum cycle length is . Hence, by assumption,

Thus is a smaller counterexample, contradicting the minimality of .

We conclude that the statement of the lemma holds. a

Our main Theorem 1.4 will follow easily from the seemingly stronger Theorem 3.2 and Corollary 3.3 below.

Theorem 3.2

Let be even, and assume there exists a -cycle decomposition of . Then, for all there exist a graph of order and a function with the following properties:

(P1)

is -partite;

(P2)

for each part of ;

(P3)

for each pair of vertices from distinct parts of ;

(P4)

admits a -edge-colouring such that, for each each colour :

(P4a)

colour class has edges;

(P4b)

for each ; and

(P4c)

has a unique non-trivial connected component.

Proof. We prove the theorem by induction on .

First we prove the basis of induction, case . Let and for all . Then the graph is of order , and Properties (P1)–(P3) clearly hold for and the function . By assumption, there exists a decomposition of into cycles of lengths . Replacing each edge in this decomposition by parallel edges we obtain a decomposition of into -fold cycles of lengths . Now define a -edge-colouring of by taking the colour class to be the -fold cycle of length in this decomposition, together with the remaining isolated vertices. Clearly, Property (P4) then holds for and as well.

Suppose now that for some there exist a graph of order and a function satisfying properties (P1)–(P4) from the statement of the theorem. We shall now construct a graph of order and a function satisfying Properties (P1)–(P4). Since and (P1)–(P2) hold for , there exists a vertex of with . The graph will be constructed as an -detachment of with the help of an auxiliary bipartite graph defined as follows.

First, define sets , , and , and let be the bipartite graph with bipartition and with for each and . Observe that, by the induction hypothesis, and for all and such that and are from distinct parts of .

By Theorem 2.2, there exists an equitable -edge-colouring of . With respect to such a colouring we have and for all , , and , where and are from distinct parts of . In particular, observe that is constant with respect to parameter (namely, it is 0 if , and 2 if ). We shall use one colour class of this equitable -edge-colouring of to define the -detachment of , however, to guarantee Property (P4c), we may need to first modify the colouring as follows.

Let be a spanning subgraph of that is the union of two arbitrary colour classes of with respect to our equitable -edge-colouring. Then and for all and such that and are from distinct parts of .

Let be the subset of containing all colours such that

(4)

We form a new (bipartite) graph from by splitting each vertex , for , into vertices and , and then divide the edges incident with so that . Theorem 2.2 gives existence of an equitable -edge-colouring of . Take an arbitrary colour class in this colouring of , and obtain a new graph from this colour class by identifying vertices and for each ; call the new vertex . Observe that for all , while for all and for all such that and are from distinct parts of .

We are now ready to define the new graph . Informally speaking, is obtained from by splitting the vertex into vertices and , and converting all edges of the form that correspond to edges of to edges of the form , preserving the colour of each edge. More formally, take any , and define as a -edge-coloured graph with and, for all and ,

Clearly, is of order and is -partite (with and in the same part). Moreover, for all , so Properties (P1) and (P4a) hold for .

We define the function as follows: , and for all . We then immediately obtain for each part of , so Property (P2) holds for and as well.

To verify Property (P4b), take any . Observe that for each . Furthermore, since , we have and .

To verify Property (P3), first observe that for any that belong to distinct parts of , and hence to distinct parts of , we have . Furthermore, for any not in the same part as and , we have , and .

It remains to verify Property (P4c), namely, that every colour class has a unique non-trivial connected component. Fix a colour . If is an isolated vertex in , then was obtained from by adjoining a new isolated vertex ; hence has a unique non-trivial connected component since does.

Hence assume is a vertex in , the unique non-trivial connected component of . Let be the subgraph of induced by . It suffices to show that is connected; since inherited all isolated vertices of , it will then follow that is the unique non-trivial connected component of