5.3 Rainbow matchings
5.3.3 Proof of the main result
Now we are able to finish the induction. By Claim 5.9, the multigraph G0 satisfies the hypothesis of Lemma 5.7. Therefore, sincen0 < n, we can apply induction, which yields that G0 contains a rainbow matchingM00of sizen0. Now,M00∪W∪e(σ) forms a rainbow matching of sizen, which is a contradiction to our assumption that there were no rainbow matchings inGof this size.
5.3.3 Proof of the main result
With Lemma 5.7 in hand, we can now present the proof of Theorem 5.4, the idea of which is as follows. If a largest rainbow matchingM in the given graph Ghas size less thann, we choose any colour c0 that is not in M and consider a subgraph G0 of G that is induced by the edges that are coloured by the colours of M or by c0. The goal is then to show that G0 contains a rainbow matching that uses all colours ofG0, which would lead to a contradiction to the maximality of M. Using the assumption on the number of vertices of the induced subgraph of each colour class allows us to show that the technical requirements of Lemma 5.7 are fulfilled. This is the place where we need to have for every colour 3n+o(n) rather than 3n−2 such vertices. Finally, Lemma 5.7 guarantees that there exists a rainbow matching using all colours inG0.
Proof of Theorem 5.4. Letδ >0,n≥144/δ2, and letGbe as stated in the theorem. For the sake of contradiction, let us assume that a largest rainbow matchingM inGhas size smaller thann. Let C0 be the set of colours in M plus one further colour c0. Set n0 := |C0|. In the following, we consider the multigraphG0 = (V, E0) with E0 ={e∈E :c(e) ∈C0}. We now apply Lemma 5.7 toG0 in order to find a rainbow matching of sizen0. This is a contradiction since we assumed thatM was a maximum matching.
Let δ0 =δn/n0 and observe that from n≥144/δ2, we have δ0 ≥12/√
n0. Let c∈C0 and let σ be any (c0, c)-switching in G0 with respect toM. By assumption on G, the number of vertices inV \ V(M)∪V(σ)
that are incident to colourcis at least
d(3 +δ)ne − |V(M)∪V(σ)|>d(1 +δ)ne −`(σ)≥ d(1 +δ0)n0e −`(σ).
If in the subgraph induced by the colour class ofc any two of these vertices are adjacent or have a common neighbour, then, since all the colour classes inGare unions of cliques, there is
an edge, saye, of colourcbetween them, which leads to the rainbow matching (M\m(σ))∪ (e(σ)∪ {e}) of sizen0. So, we may assume that there are at least d(1 +δ0)n0e −`(σ)
disjoint edges of colour c inEG0
V \ V(M)∪V(σ) , V
. Therefore and since there can be at most 3`(σ) disjointc-edges inEG0
V\ V(M)∪V(σ)
, V(σ)
, there are at least d(1+δ0)n0e−4`(σ) disjoint c-edges inEG0
V \ V(M)∪V(σ)
, V \V(σ)
. As c and σ were chosen arbitrarily, Lemma 5.7 now guarantees thatG0 contains a rainbow matching of sizen0.
5.4 Concluding remarks
We wonder how Theorem 5.3 changes if one adds the natural constraint that every pair of distinct elements belongs to at most one equivalence relation. More precisely, we are interested in the following problem.
Problem 5.10. Determine the minimal numberv∗(n)such that ifA1, . . . , Anare equivalence relations on a setX with |ker(Ai)| ≥v∗(n) and Ai∩Aj =
(x, x) : x ∈X for all distinct indicesi, j∈[n], then A1, . . . , An contains a rainbow matching.
Using the graph theoretic notion as before, the additional constraint means that the colour classes are pairwise disjoint. This can also be seen as restricting the problem to graphs instead of considering multigraphs. It is known that for every evenn, there exists an edge-coloured bipartite graph whose colour classes induce matchings of size n and that does not contain a rainbow matching of sizen. This follows from the fact that for every even n there exists a Latin square of order n without a transversal (see Subsection 1.2.2). For general n we thus obtain v∗(n) > 2n−2. An upper bound on v∗(n) follows directly from Theorem 5.4, i.e.v∗(n)≤v(n) = 3n+o(n).
Corollary 5.5 assures that a collection of nmatchings of size (3/2 +o(1))nin a bipartite multigraph guarantees a rainbow matching of size n. Aharoni, Kotlar, and Ziv [5] slightly improved the lower bound on the sizes of the matchings tod3n/2e+ 1. For smaller matching sizes, it is unknown whether a rainbow matching of size n−1 exists. More generally, as suggested by Tibor Szab´o (private communication), it would be interesting to determine upper bounds on the smallest integer µ(n, `) such that every family of n matchings of size µ(n, `) in a bipartite multigraph guarantees a rainbow matching of size n−`. One can verify that µ(n, l) ≤ l+2l+1n. Moreover, it holds that µ(n,√
n) ≤n, which is a generalisation (see e.g. [4, 19]) of a result proved in the context of Latin squares by Woolbright [158], and independently by Brouwer, de Vries and Wieringa [45].
In order to approach Conjecture 1.13, one can also increase the number of matchings and fix their sizes to be equal to n instead of considering families of n matchings of sizes greater than n. Drisko [65] proved that a collection of 2n−1 matchings of size n in a bipartite multigraph with partition classes of sizenguarantees a rainbow matching of size n.
This result is tight, as the factorisation of a cycle on 2n vertices with edges of multiplicity n−1 show. The problem was further investigated in the following two directions. Does the statement also hold if we omit the restriction on the sizes of the vertex classes? And how many matchings do we need to find a rainbow matching of sizen−` for every`≥1?
Aharoni and Berger [2] affirmed the first question by showing that for any two integers s≤t, the maximal number of matchings of sizetin a bipartite multigraph that do not contain a rainbow matching of sizes is equal to 2(s−1).
5.4. Concluding remarks 145
The second question was studied recently by Bar´at, Gy´arf´as, and S´ark¨ozy in [19]. They proved that for every`≥1 any bipartite multigraph withj
`+2
`+1nk
−(`+ 1) matchings of sizen has a rainbow matching of sizen−`. This result is best possible for`= 0 andbn/2c ≤` < n.
Finally, if Conjecture 1.13 is true, it is of interest to see how tight it is. As shown by Bar´at and Wanless [20], one can find constructions ofn matchings with n
2
−1 matchings of size n+ 1 and the remaining ones being of size n such that there is no rainbow matching of size n. We wonder whether the expression n
2
−1 above could also be replaced by (1−o(1))n.
Enumerating spanning trees in series-parallel graphs 6
While in the preceding chapters we were examining questions concerning the existence of certain substructures, in this chapter we are interested in thenumber of spanning subgraphs.
As motivated and summarised in Subsection 1.2.3, the enumeration of spanning trees and the analysis of series-parallel graphs (or SP graphs for short) have a rich history. By means of analytic techniques we show that the expected number of spanning trees in a connected labelled SP graph onnvertices chosen uniformly at random satisfies an estimate of the form
s%−n 1 +o(1) ,
wheres≈0.09063 and%−1≈2.08415 can be computed explicitly (Theorem 6.1). We obtain analogue results for subfamilies of SP graphs including 2-connected SP graphs (Theorem 6.1), 2-trees (Theorem 6.8), and SP graphs with fixed excess (Theorem 6.12). Our proofs are based on analytic combinatorics, especially on the symbolic method and the singularity analysis of generating functions. The necessary analytic background was introduced in Section 2.3.
Since we focus on enumerative problems defined on SP graphs, let us quickly state the following alternative definition of SP graphs, which provides more insight into their structure and also justifies their name: let G be a graph and let s and t be two of its vertices. We say G is series-parallel with terminals s and t ifG can be turned into the single edge {s, t} by a sequence of the following operations: replacement of a pair of parallel edges (i.e. edges sharing two common endpoints) by a single edge, or replacement of a pair of series edges (i.e. non-parallel edges sharing a common endpoint of degree 2) by a single edge. A graph G is 2-terminal series-parallel if there exist vertices s and t inG such that Gis series-parallel with terminalssandt. Finally, a graphGisseries-parallel if and only if each of its 2-connected components is a 2-terminal series-parallel graph (see e.g. [44]).
This chapter is structured in the following way. In Section 6.1 we prove Theorem 6.1 and analyse the behaviour of the growth constant of the expected number of spanning trees if we fix the edge density of a random 2-connected SP graph. We also comment on the variance of the number of spanning trees in a 2-connected SP graph chosen uniformly at random. Next, Section 6.2 is devoted to the analysis of the number of spanning trees in edge-maximal SP graphs (Theorem 6.8). The proof of Theorem 6.12, which deals with connected SP graphs with fixed excess, is presented in Section 6.3. We close the chapter with some concluding remarks and open questions in Section 6.4.
147
Throughout this chapter all graphs under study are labelled, unless stated otherwise.
Furthermore, in contrast to the previous chapters, by a random object of a given family we mean an object chosen uniformly at random from all the elements of the same size, e.g. graphs on the same number of vertices. As already mentioned, this chapter is based on joint work with Juanjo Ru´e [75].
6.1 Spanning trees in connected and 2-connected SP graphs
The goal of this section is to analyse the expected value and the variance of spanning trees in random connected and 2-connected SP graphs as well as to elaborate on the growth constant of the expected number of spanning trees in random 2-connected SP graphs of a given edge density. Our main result in this respect is the following theorem. Its proof is presented in Subsection 6.1.1.
Theorem 6.1. Let Xn and Zn denote the number of spanning trees in a connected, respec-tively 2-connected labelled SP graph onn vertices chosen uniformly at random. Then
E[Xn]= s%−n 1 +o(1)
, where s≈0.09063, %−1 ≈2.08415, E[Zn] =p$−n 1 +o(1)
, where p≈0.25975, $−1≈2.25829.
Since the number of connected/2-connected SP graphs was already determined asymp-totically by Bodirsky, Gim´enez, Kang, and Noy [33] (see Theorem 1.14), we may reduce the problem of estimating the number of spanning trees in random connected/2-connected SP graphs to the enumeration of connected/2-connected SP graphs carrying a distinguished spanning tree.
For this purpose, let cn,m and bn,m denote the number of connected and 2-connected SP graphs with a distinguished spanning tree, respectively. LetC(x, y) andB(x, y) be their as-sociated counting formula, wherexandymark vertices and edges, respectively. Furthermore, let Ddenote the class of series-parallel networks carrying a distinguished spanning tree and let D(x, y) be its associated generating function.