The chromatic index of a graph , denoted by , is the minimum number of colors with which it is possible to color the edges of
in a way such that every color class consists of a matching (that is, no two edges of the same color share a vertex). This parameter is one of the most fundamental and widely studied parameters in graph theory and combinatorial optimization and in particular, is related to optimal scheduling and resource allocation problems and round-robin tournaments (see, e.g.,, , ).
A trivial lower bound on is , where denotes the maximum degree of . Indeed, consider any vertex with maximum degree, and observe that all edges incident to this vertex must have distinct colors. Perhaps surprisingly, a classical theorem of Vizing  from the 1960s shows that colors are always sufficient, and therefore, holds for all graphs. In particular, this shows that one can partition all graphs into two classes: Class 1 consists of all graphs for which , and Class 2 consists of all graphs for which . Moreover, the strategy in Vizing’s original proof can be used to obtain a polynomial time algorithm to edge color any graph with colors (). However, Holyer  showed that it is actually NP-hard to decide whether a given graph is in Class 1 or 2. In fact, Leven and Galil  showed that this is true even if we restrict ourselves to graphs with all the degrees being the same (that is, to regular graphs).
Note that for -regular graphs (that is, graphs with all their degrees equal to ) on an even number of vertices, the statement ‘ is of Class 1’ is equivalent to the statement that contains edge-disjoint perfect matchings (also known as -factors). A graph whose edge set decomposes as a disjoint union of perfect matchings is said to admit a -factorization. Note that if is a -regular bipartite graph, then a straightforward application of Hall’s marriage theorem immediately shows that is of Class 1. Unfortunately, the problem is much harder for non-bipartite graphs, and it is already very interesting to find (efficiently verifiable) sufficient conditions which ensure that . This problem is the main focus of our paper.
1.1 Regular expanders are of Class 1
Our main result shows that -regular graphs on an even number of vertices which are ‘sufficiently good’ spectral expanders, are of Class 1. Before stating our result precisely, we need to introduce some notation and definitions. Given a -regular graph on vertices, let be its adjacency matrix (that is, is an , -valued matrix, with if and only if ). Clearly, , where
is the vector with all entries equal to, and therefore, is an eigenvalue of . In fact, as can be easily proven, is the eigenvalue of with largest absolute value. Moreover, since is a symmetric, real-valued matrix, it has real eigenvalues (counted with multiplicities). Let
denote the eigenvalues of , and let . With this notation, we say that is an -graph if is a -regular graph on vertices with . In recent decades, the study of graphs, also known as ‘spectral expanders’, has attracted considerable attention in mathematics and theoretical computer science. An example which is relevant to our problem is that of finding a perfect matching in -graphs. Extending a result of Krivelevich and Sudakov , Ciobǎ, Gregory and Haemers  proved accurate spectral conditions for an -graph to contain a perfect matching. For much more on these graphs and their many applications, we refer the reader to the surveys of Hoory, Linial and Wigderson , Krivelevich and Sudakov , and to the book of Brouwer and Haemers . We are now ready to state our main result.
For every there exist such that for all even integers and for all the following holds. Suppose that is an -graph with . Then, .
It seems plausible that with a more careful analysis of our proof, one can improve our bound to . Since we believe that the actual bound should be much stronger, we did not see any reason to optimize our bound at the expense of making the paper more technical.
In particular, since the eigenvalues of a matrix can be computed in polynomial time, Theorem 1.1 provides a polynomial time checkable sufficient condition for a graph to be of Class 1. Moreover, our proof gives a probabilistic polynomial time algorithm to actually find an edge coloring of such a with colors. Our result can be viewed as implying that ‘sufficiently good’ spectral expanders are easy instances for the NP-complete problem of determining the chromatic index of regular graphs. It is interesting (although, perhaps a bit unrelated) to note that in recent work, Arora et al.  showed that constraint graphs which are reasonably good spectral expanders are easy for the conjecturally NP-complete Unique Games problem as well.
1.2 Almost all -regular graphs are of Class 1
The phrase ‘almost all -regular graphs’ usually splits into two cases: ‘dense’ graphs and random graphs. Let us start with the former.
Dense graphs: It is well known (and quite simple to prove) that every -regular graph on vertices, with has a perfect matching (assuming, of course, that is even). Moreover, for every , it is easily seen that there exist -regular graphs on an even number of vertices that do not contain even one perfect matching. In a (relatively) recent breakthrough, Csaba, Kühn, Lo, Osthus, and Treglown  proved a longstanding conjecture of Dirac from the 1950s, and showed that the above minimum degree condition is tight, not just for containing a single perfect matching, but also for admitting a -factorization.
Theorem 1.3 (Theorem 1.1.1 in ).
Let be a sufficiently large even integer, and let . Then, every -regular graph on vertices contains a -factorization.
Hence, every sufficiently ‘dense’ regular graph is of Class 1. It is worth mentioning that they actually proved a much more general statement about finding edge-disjoint Hamilton cycles, from which the above theorem follows as a corollary.
Random graphs: As noted above, one cannot obtain a statement like Theorem 1.3 for smaller values of since the graph might not even have a single perfect matching. Therefore, a natural candidate to consider for such values of is the random -regular graph, denoted by
, which is simply a random variable that outputs a-regular graph on vertices, chosen uniformly at random from among all such graphs. The study of this random graph model has received much interest in recent years. Unlike the traditional binomial random graph
(where each edge of the complete graph is included independently, with probability), the uniform regular model has many dependencies, and is therefore much harder to work with. For a detailed discussion of this model, along with many results and open problems, we refer the reader to the survey of Wormald .
Working with this model, Janson , and independently, Molloy, Robalewska, Robinson, and Wormald , proved that a typical admits a -factorization for all fixed , where is a sufficiently large (depending on ) even integer. Later, Kim and Wormald  gave a randomized algorithm to decompose a typical into edge-disjoint Hamilton cycles (and an additional perfect matching if
is odd) under the same assumption thatis fixed, and is a sufficiently large (depending on ) even integer. The main problem with handling values of which grow with is that the so-called ‘configuration model’ (see  for more details) does not help us in this regime.
Here, as an almost immediate corollary of Theorem 1.1, we deduce the following, which together with the results of  and  shows that a typical on a sufficiently large even number of vertices admits a -factorization for all .
There exists a universal constant such that for all , a random -regular graph admits a -factorization asymptotically almost surely (a.a.s.).
By asymptotically almost surely, we mean with probability going to as goes to infinity (through even integers). Since a -factorization can never exist when is odd, we will henceforth always assume that is even, even if we do not explicitly state it.
To deduce Corollary 1.4 from Theorem 1.1, it suffices to show that we have (say) a.a.s. In fact, the considerably stronger (and optimal up to the choice of constant in the big-oh) bound that a.a.s. is known. For , this is due to Broder, Frieze, Suen and Upfal . This result was extended to the range by Cook, Goldstein, and Johnson  and to all values of by Tikhomirov and Youssef . We emphasize that the condition on we require is significantly weaker and can possibly be deduced from much simpler arguments than the ones in the references above.
It is also worth mentioning that very recently, Haxell, Krivelevich and Kronenberg  studied a related problem in a random multigraph setting; it is interesting to check whether our techniques can be applied there as well.
1.3 Counting -factorizations
Once the existence of -factorizations in a family of graphs has been established, it is natural to ask for the number of distinct -factorizations that any member of such a family admits. Having a ‘good’ approximation to the number of -factorizations can shed some light on, for example, properties of a ‘randomly selected’ -factorization. We remark that the case of counting the number of -factors (perfect matchings), even for bipartite graphs, has been the subject of fundamental works over the years, both in combinatorics (e.g., , , , ), as well as in theoretical computer science (e.g., , ), and had led to many interesting results such as both closed-form as well as computational approximation results for the permanent of matrices.
As far as the question of counting the number of -factorizations is concerned, much less is known. Note that for
-regular bipartite graphs, one can use estimates on the permanent of the adjacency matrix ofto obtain quite tight results. But quite embarrassingly, for non-bipartite graphs (even for the complete graph!) the number of -factorizations in unknown. The best known upper bound for the number of -factorizations in the complete graph is due to Linial and Luria , who showed that it is upper bounded by
Moreover, by following their argument verbatim, one can easily show that the number of 1-factorizations of any -regular graph is at most
which is off by a factor of from the upper bound.
An immediate advantage of our proof is that it gives a lower bound on the number of -factorizations which is better than the one above by a factor of in the base of the exponent, not just for the complete graph, but for all sufficiently good regular spectral expanders with degree greater than some large constant. More precisely, we will show the following (see also the third bullet in Section 7)
For any , there exist such that for all even integers and for all , the number of -factorizations in any -graph with is at least
As discussed before, this immediately implies that for all , the number of -factorizations of is a.a.s.
1.4 Outline of the proof
It is well known, and easily deduced from Hall’s theorem, that any regular bipartite graph admits a -factorization (Corollary 2.9). Therefore, if we had a decomposition , where are regular balanced bipartite graphs, and is a -factorization of the regular graph , we would be done. Our proof of Theorem 1.1 will obtain such a decomposition constructively.
As shown in Proposition 5.1, one can find a collection of edge disjoint, regular bipartite graphs , where and each is regular, with , which covers ‘almost’ all of . In particular, one can find an ‘almost’ -factorization of . However, it is not clear how to complete an arbitrary such ‘almost’ -factorization to an actual -factorization of . To circumvent this difficulty, we will adopt the following strategy. Note that is a -regular graph with , and we can further force to be even (for instance, by removing a perfect matching from ). Therefore, by Petersen’s -factor theorem (Theorem 2.13), we easily obtain a decomposition , where each is approximately regular. The key ingredient of our proof (Proposition 4.2) then shows that the ’s can initially be chosen in such a way that each can be edge decomposed into a regular balanced bipartite graph, and a relatively small number of -factors.
The basic idea in this step is quite simple. Observe that while the regular graph is not bipartite, it is ‘close’ to being one, in the sense that most of its edges come from the regular balanced bipartite graph . Let denote the graph induced by on the vertex set , and similarly for , and note that the number of edges . We will show that can be taken to have a certain ‘goodness’ property (Definition 4.1) which, along with the sparsity of , enables one to perform the following process to ‘absorb’ the edges in and : decompose and into the same number of matchings, with corresponding matchings of equal size, and complete each such pair of matchings to a perfect matching of . After removing all the perfect matchings of obtained in this manner, we are clearly left with a regular balanced bipartite graph, as desired.
2 Tools and auxiliary results
In this section we have collected a number of tools and auxiliary results to be used in proving our main theorem.
2.1 Probabilistic tools
Throughout the paper, we will make extensive use of the following well-known concentration inequality due to Hoeffding ().
Lemma 2.1 (Hoeffding’s inequality).
Let be independent random variables such that with probability one. If , then for all ,
Sometimes, we will find it more convenient to use the following bound on the upper and lower tails of the Binomial distribution due to Chernoff (see, e.g., Appendix A in).
Lemma 2.2 (Chernoff’s inequality).
Let and let . Then
for every ;
for every .
These bounds also hold when
is hypergeometrically distributed with mean.
Before introducing the next tool to be used, we need the following definition.
Let be a collection of events in some probability space. A graph on the vertex set is called a dependency graph for if is mutually independent of all the events .
The following is the so called Lovász Local Lemma, in its symmetric version (see, e.g., ).
Lemma 2.5 (Local Lemma).
Let be a sequence of events in some probability space, and let be a dependency graph for . Let and suppose that for every we have , such that . Then, .
We will also make use of the following asymmetric version of the Lovász Local Lemma (see, e.g., ).
Lemma 2.6 (Asymmetric Local Lemma).
Let be a sequence of events in some probability space. Suppose that is a dependency graph for , and suppose that there are real numbers , such that and
for all . Then, .
2.2 Perfect matchings in bipartite graphs
Here, we present a number of results related to perfect matchings in bipartite graphs. The first result is a slight reformulation of the classic Hall’s marriage theorem (see, e.g., ).
Let be a balanced bipartite graph with . Suppose for all subsets of size at most which are completely contained either in or in . Then, contains a perfect matching.
Moreover, we can always find a maximum matching in a bipartite graph in polynomial time using standard network flow algorithms (see, e.g., ).
The following simple corollaries of Hall’s theorem will be useful for us.
Every -regular balanced bipartite graph has a perfect matching, provided that .
Let be an -regular graph. Let be a set of size at most . Note that as is -regular, we have
Since each vertex in has degree at most into , we get
Similarly, for every of size at most we obtain
Therefore, by Theorem 2.7, we conclude that contains a perfect matching. ∎
Since removing an arbitrary perfect matching from a regular balanced bipartite graph leads to another regular balanced bipartite graph, a simple repeated application of Corollary 2.8 shows the following:
Every regular balanced bipartite graph has a -factorization.
In fact, as the following theorem due to Schrijver  shows, a regular balanced bipartite graph has many 1-factorizations.
The number of -factorizations of a -regular bipartite graph with vertices is at least
The next result is a criterion for the existence of -factors (that is, -regular, spanning subgraphs) in bipartite graphs, which follows from a generalization of the Gale-Ryser theorem due to Mirsky .
Let be a balanced bipartite graph with , and let be an integer. Then, contains an -factor if and only if for all and
Moreover, such factors can be found efficiently using standard network flow algorithms (see, e.g., ).
As we are going to work with pseudorandom graphs, it will be convenient for us to isolate some ‘nice’ properties that, together with Theorem 2.11, ensure the existence of large factors in balanced bipartite graphs.
Let be a balanced bipartite graph with . Suppose there exist and satisfying the following additional properties:
for all .
for all and with .
for all and with and .
Then, contains an -factor.
By Theorem 2.11, it suffices to verify that for all and we have
We divide the analysis into five cases:
. In this case, we trivially have
so there is nothing to prove.
and . Since , we always have . Thus, it suffices to verify that
Assume, for the sake of contradiction, that this is not the case. Then, since there are at least edges incident to , we must have
However, since , this contradicts .
and . This is exactly the same as the previous case with the roles of and interchanged.
, and . By , it suffices to verify that
Dividing both sides by , the above inequality is equivalent to
where , , , , and .
Since by , this is readily verified on the (triangular) boundary of the region, on which the inequality reduces to one of the following: ; ; . On the other hand, the only critical point in the interior of the region is possibly at , for which we have , again by .
, and . This is exactly the same as the previous case with the roles of and interchanged.
2.3 Matchings in graphs with controlled degrees
In this section, we collect a couple of results on matchings in (not necessarily bipartite) graphs satisfying some degree conditions. A -factorization of a graph is a decomposition of its edges into -factors (that is, a collection of vertex disjoint cycles that covers all the vertices). The following theorem, due to Petersen , is one of the earliest results in graph theory.
Theorem 2.13 (2-factor Theorem).
Every -regular graph with admits a 2-factorization.
The next theorem, due to Vizing , shows that every graph admits a proper edge coloring using at most colors.
Theorem 2.14 (Vizing’s Theorem).
Every graph with maximum degree can be properly edge-colored with colors.
2.4 Expander Mixing Lemma
When dealing with graphs, we will repeatedly use the following lemma (see, e.g., ), which bounds the difference between the actual number of edges between two sets of vertices, and the number of edges we expect based on the sizes of the sets.
Lemma 2.15 (Expander Mixing Lemma).
Let be an graph, and let . Let . Then,
3 Random partitioning
While we have quite a few easy-to-use tools for working with balanced bipartite graphs, the graph we start with is not necessarily bipartite (when the starting graph is bipartite, the existence problem is easy, and the counting problem is solved by ). Therefore, perhaps the most natural thing to do is to partition the edges into ‘many’ balanced bipartite graphs, where each piece has suitable expansion and regularity properties. The following lemma is our first step towards obtaining such a partition.
Fix , and let be a -regular graph on vertices, where and is an even and sufficiently large integer. Then, for every integer , there exists a collection of balanced bipartitions for which the following properties hold:
Let be the subgraph of induced by . For all we have
For all , the number of indices for which is .
We will divide the proof into two cases – the dense case, where , and the sparse case, where . The underlying idea is similar in both cases, but the proof in the sparse case is technically more involved as a standard use of Chernoff’s bounds and the union bound does not work (and therefore, we will instead use the Local Lemma).
Proof in the dense case.
be random subsets chosen independently from the uniform distribution on all subsets ofof size exactly , and let for all . We will show that with high probability, for every , is a balanced bipartition satisfying and .
First, note that for any and any ,
Therefore, if for all we let denote the set of indices for which , then
Next, note that, for a fixed , the events ’ are mutually independent, and that , where
is the indicator random variable for the event. Therefore, by Chernoff’s bounds, it follows that
Now, by applying the union bound over all , it follows that the collection satisfies with probability at least . Similarly, it is immediate from Chernoff’s inequality for the hypergeometric distribution that for any and ,
and by taking the union bound over all such and , it follows that holds with probability at least . All in all, with probability at least , both and hold. This completes the proof. ∎
Proof in the sparse case.
Instead of using the union bound as in the dense case, we will use the symmetric version of the Local Lemma (Lemma 2.5). Note that there is a small obstacle with choosing balanced bipartitions, as the local lemma is most convenient to work with when the underlying experiment is based on independent trials. In order to overcome this issue, we start by defining an auxiliary graph as follows: for all , if and only if and there is no vertex with . In other words, there is an edge between and in if and only if and are not connected to each other, and do not have any common neighbors in . Since for any , there are at most many such that or and have a common neighbor in , it follows that for sufficiently large. An immediate application of Hall’s theorem shows that any graph on vertices with minimum degree at least contains a perfect matching. Therefore, contains a perfect matching.
Let and let be an arbitrary perfect matching of . For each let be a random function chosen independently and uniformly from the set of all functions from to . These functions will define the partitions according to the vertex labels as follows:
Clearly, is a random collection of balanced bipartitions of . If, for all , we let denote the random function recording which of or a given vertex ends up in, it is clear – and this is the point of using – that for any and any , the choices are mutually independent. This will help us in showing that, with positive probability, this collection of bipartitions satisfies properties and .
Indeed, for all , , and , let denote the event that ‘’, and let denote the event ‘’. Then, using the independence property mentioned above, Chernoff’s bounds imply that
In order to complete the proof, we need to show that one can apply the symmetric local lemma (Lemma 2.5) to the collection of events consisting of all the ’s and all the ’s. To this end, we first need to upper bound the number of events which depend on any given event.
Note that depends on only if and or . Note also that depends on only if an end point of is within distance of either in or in . Therefore, it follows that any can depend on at most events in the collection. Since can depend on only if and share an endpoint in or if any of the endpoints of are matched to any of the endpoints in , it follows that we can take the maximum degree of the dependency graph in Lemma 2.5 to be . Since , we are done. ∎
In this section, we describe the key ingredient of our proof, namely the completion step. Before stating the relevant lemma, we need the following definition.
A graph is called -good if it satisfies the following properties:
is an -regular, balanced bipartite graph with .
Every balanced bipartite subgraph of with and with contains a perfect matching.
The motivation for this definition comes from the next proposition, which shows that a regular graph on an even number of vertices, which can be decomposed as a union of a good graph and a sufficiently sparse graph, has a 1-factorization.
For every , there exists an integer such that for all and a sufficiently large integer, the following holds. Suppose that is an -good graph. Then, for every , every -regular (not necessarily bipartite) graph on the vertex set , for which , admits a 1-factorization.
For clarity of exposition, we first provide the somewhat simpler proof (which already contains all the ideas) of this proposition in the ‘dense’ case, and then we proceed to the ‘sparse’ case.
Proof in the dense case: .
First, observe that . Indeed, as is -regular, we have for that
from which the above equality follows. Moreover, for all . Next, let and . By Vizing’s theorem (Theorem 2.14), both and contain matchings of size exactly . Consider any two such matchings in and in , and for , let , denote a matching of size such that no vertex is incident to more than vertices which are paired in the matching. To see that such an must exist, we use a simple probabilistic argument – for a random subset of this size, by a simple application of Hoeffding’s inequality and the union bound, we obtain that satisfies the desired property, except with probability at most .
Delete the vertices in , as well as any edges incident to them, from and denote the resulting graph by . Since and by the choice of , it follows from that contains a perfect matching . Note that is a perfect matching in . We repeat this process with (deleting only the edges in , and not the vertices) and until we reach and such that . Since , this must happen after at most steps. Moreover, since , it follows that during the first steps of this process, the degree of any is at least . Therefore, since , we can indeed use throughout the process, as done above.
From this point onwards, we continue the above process (starting with ) with matchings of size one i.e. single edges from each part, until no more edges are left. By the choice of , we need at most such iterations, which is certainly possible since . After removing all the perfect matchings obtained via this procedure, we are left with a regular, balanced, bipartite graph, and therefore it admits a 1-factorization (Corollary 2.9). Taking any such 1-factorization along with all the perfect matchings that we removed gives a 1-factorization of . ∎
Proof in the sparse case: .
Let be any integer and let . We begin by showing that any matching in with can be partitioned into matchings such that no vertex is incident to more than vertices in for any . If , then there is nothing to show. If , consider an arbitrary partition of into sets with each set (except possibly the last one) of size . For each , , choose a permutation of independently and uniformly at random, and let denote the random subset of consisting of all elements of which are assigned the label . We will show, using the symmetric version of the Local Lemma (Lemma 2.5), that the decomposition satisfies the desired property with a positive probability.
To this end, note that for any vertex to have at least neighbors in some , it must be the case that the neighbors of in are spread throughout at least distinct ’s. Let denote the event that has at least neighbors in some matching . Since has at least neighbors in at most distinct ’s, it follows that . Finally, since each depends on at most many other ’s, and since , we are done.
Now, as in the proof of the dense case, we have and for . By Vizing’s theorem, we can decompose and into exactly matchings each, and it is readily seen that these matchings can be used to decompose and into at most matchings and such that for all . Using the argument in the previous paragraph, we can further decompose each , into at most matchings each such that no vertex is incident to more than vertices involved in any of these smaller matchings. Since , the rest of the argument proceeds exactly as in the dense case. ∎
In the last step of the proof, we are allowed to choose an arbitrary -factorization of an -regular, balanced bipartite graph, where . Therefore, using Theorem 2.10 along with the standard inequality , it follows that admits at least -factorizations.
5 Finding good subgraphs which almost cover
In this section we present a structural result which shows that a ‘good’ regular expander on an even number of vertices can be ‘almost’ covered by a union of edge disjoint good subgraphs.
For every there exists such that for all the following holds. Let be an -graph with where is an integer in . Then, contains distinct, edge disjoint -good subgraphs with and .
There exists an edge partitioning for which the following properties hold:
is a balanced bipartite graph with parts for all .
For all and for all , with we have
For all and all , with and ,