Scholarly article on topic 'Existences of rainbow matchings and rainbow matching covers'

Existences of rainbow matchings and rainbow matching covers Academic research paper on "Computer and information sciences"

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Abstract of research paper on Computer and information sciences, author of scientific article — Allan Lo

Abstract Let G be an edge-coloured graph. A rainbow subgraph in G is a subgraph such that its edges have distinct colours. The minimum colour degree δ c ( G ) of G is the smallest number of distinct colours on the edges incident with a vertex of  G . We show that every edge-coloured graph G on n ≥ 7 k / 2 + 2 vertices with δ c ( G ) ≥ k contains a rainbow matching of size at least k , which improves the previous result for k ≥ 10 . Let Δ mon ( G ) be the maximum number of edges of the same colour incident with a vertex of G . We also prove that if t ≥ 11 and Δ mon ( G ) ≤ t , then G can be edge-decomposed into at most ⌊ t n / 2 ⌋ rainbow matchings. This result is sharp and improves a result of LeSaulnier and West.

Academic research paper on topic "Existences of rainbow matchings and rainbow matching covers"

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Discrete Mathematics

journal homepage: www.elsevier.com/locate/disc

Existences of rainbow matchings and rainbow matching covers

Allan Lo

School of Mathematics, University of Birmingham, Birmingham, B152TT, United Kingdom

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Article history:

Received 24 March 2014

Received in revised form 12 May 2015

Accepted 15 May 2015

Available online 9 June 2015

Keywords: Edge coloring Rainbow Matching

abstract

Let G be an edge-coloured graph. A rainbow subgraph in G is a subgraph such that its edges have distinct colours. The minimum colour degree 5c (G) of G is the smallest number of distinct colours on the edges incident with a vertex of G. We show that every edge-coloured graph G on n > 7k/2 + 2 vertices with 5c (G) > k contains a rainbow matching of size at least k, which improves the previous result for k > 10.

Let 4mon (G) be the maximum number of edges of the same colour incident with a vertex of G. We also prove that if t > 11 and 4mon(G) < t, then G can be edge-decomposed into at most |_tn/2J rainbow matchings. This result is sharp and improves a result of LeSaulnier and West.

© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Let G be a simple graph, that is, it has no loops or multi-edges. We write V (G) for the vertex set of G and 5(G) for the minimum degree of G. An edge-coloured graph is a graph in which each edge is assigned a colour. We say that an edge-coloured graph G is proper if no two adjacent edges have the same colour. A subgraph H of G is rainbow if all its edges have distinct colours. Rainbow subgraphs are also called totally multicoloured, polychromatic, or heterochromatic subgraphs.

In this paper, we are interested in rainbow matchings in edge-coloured graphs. The study of rainbow matchings began with a conjecture of Ryser [10], which states that every Latin square of odd order contains a Latin transversal. Equivalently, for n odd, every properly n-edge-colouring of Knn, the complete bipartite graph with n vertices on each part, contains a rainbow copy of a perfect matching. In a more general setting, given a graph H, we wish to know if an edge-coloured graph G contains a rainbow copy of H. A survey on rainbow matchings and other rainbow subgraphs in edge-coloured graphs can be found in [3].

For a vertex v of an edge-coloured graph G, the colour degree, dc (v), of v is the number of distinct colours on the edges incident with v. The smallest colour degree of all vertices in G is the minimum colour degree of G and is denoted by 5c (G). Note that a properly edge-coloured graph G with 5(G) > k has 5c (G) > k.

Li and Wang [8] showed that if 5c(G) = k, then G contains a rainbow matching of size [(5k — 3)/12"|. They further conjectured that if k > 4, then G contains a rainbow matching of size [k/2|. LeSaulnier et al. [6] proved that if 5c(G) = k, then G contains a rainbow matching of size |_k/2J. The conjecture was later proved in full by Kostochka and Yancey [4].

Wang [11] asked does there exist a function f (k) such that every properly edge-coloured graph G on n > f (k) vertices with 5(G) > k contains a rainbow matching of size at least k. Diemunsch et al. [1] showed that such function does exist and f (k) < 98k/23. Gyarfas and Sarkozy [2] improved the result to f (k) < 4k — 3. Independently, Tan and the author [9] showed that f (k) < 4k — 4 for k > 4.

E-mail address: s.a.lo@bham.ac.uk. http://dx.doi.Org/10.1016/j.disc.2015.05.015

0012-365X/© 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

Kostochka, Pfender and Yancey [5] showed that every (not necessarily properly) edge-coloured G on n > 17k2/4 vertices with Sc (G) > k contains a rainbow matching of size k. Tan and the author [9] improved the bound to n > 4k — 4 for k > 4. In this paper we show that n > 7k/2 + 2 is sufficient.

Theorem 1.1. Every edge-coloured graph Gonn > 7k/2 + 2 vertices with Sc (G) > k contains a rainbow matching of size k. Moreover if G is bipartite, then we further improve the bound to n > (3 + s)k + s-2.

Theorem 1.2. Let 0 < s < 1/2 andk e N. Every edge-coloured bipartite graph G on n > (3+s)k+s—2 verticeswith Sc (G) > k contains a rainbow matching of size k.

We also consider covering an edge-coloured graph G by rainbow matchings. Given an edge-coloured graph G, let 4mon (G) be the largest maximum degree of monochromatic subgraphs of G. LeSaulnier and West [7] showed that every edge-coloured graph G on n vertices with 4mon(G) < t has an edge-decomposition into at most t(1 +1)n ln n rainbow matchings. We show that G can be edge-decomposed into |_tn/2J rainbow matchings provided t > 11.

Theorem 1.3. Forallt > 11, every edge-coloured graph G on n vertices with Amon(G) < t can be edge-decomposed into |_tn/2J rainbow matchings.

Note that the bound is best possible by considering edge-coloured graphs, where one colour class induces a t-regular graph.

Theorems 1.1 and 1.2 are proved in Section 2. Theorem 1.3 is proved in Section 3. 2. Existence of rainbow matchings

We write [k] for {1, 2,..., k}. Let G be a graph with an edge-colouring c. We denote by c(G) the set of colours in G. We write |G| for IV(G) |. Given W c V(G), G[W] is the induced subgraph of G on W. All colour sets are assumed to be finite.

Before proving Theorems 1.1 and 1.2, we consider the following (weaker) question. Suppose that G is an edge-coloured graph and contains a rainbow matching M of size k — 1. Under what colour degree and |G| conditions can we 'extend' M into a matching of size k with at least k — 1 colours? We formalise the question below.

Let G be a family of graphs closed under vertex/edge deletions. Define y(G) to be the smallest constant y such that, whenever k e N, G e G is a graph with |G| > y k and an edge-colouring c on G, the following holds. If for any rainbow matching M of size k — 1 in G, we have dc(z) > k for all z e V(G)\V(M), then G contains a rainbow matching M' of size k — 1 and a disjoint edge. (Note that the colour of the disjoint edge may appear in M'.) Clearly, y(G) > 2 for any family G of graphs. It is easy to see that equality holds if G is the family of bipartite graphs.

Proposition 2.1. Let G be the family of bipartite graphs. Then y(G) = 2.

Proof. Let G be a bipartite graph on at least 2k vertices. Suppose that M is a rainbow matching of size k — 1 and that dc (z) > k for all z e V(G)\V(M). Since G is bipartite, there exists an edge vertex-disjoint from M and so the proposition follows. □

If G is the family of all graphs, we will show that y(G) < 3.

Lemma 2.2. Let Gbea graph with at least 3(k — 1) + 1 vertices. Suppose that M is a rainbow matching of size k — 1 and that dc (z) > kfor all z e V (G)\V (M). Then G contains a rainbow matching M' of size k — 1 and a disjoint edge.

Proof. Let x1y1,..., xk— 1yk—1 be the edges of M with c(xiyi) = i. Let W = V(G)\V(M). We may assume that G[W] is empty or else the lemma holds easily.

Suppose the lemma does not hold for G. By relabelling the indices of i and swapping the roles of xi and yi if necessary, we will show that there exist distinct vertices z1,..., zk— 1 in W such that for each 1 < i < k — 1, the following holds:

(ai) yizi is an edge and c(yz) <£ [i].

(bi) Let Ti be the vertex set {xj, yj, zj : i < j < k — 1}. For any colour j', there exists a rainbow matching Mj, of size k — i on

Ti such that c(Mj,) n ([i — 1] U {/}) = 0. (ci) Let Wi = W \ {zu z+1,..., z—}. For all w e Wit N (w) n Ti £&,..., yk— 1}.

Let Wk = W and Tk = 0. Suppose that we have already found zk—1, zk—2,..., zi+1. We find zi as follows.

Note that |Wi+1| > n — 2(k — 1) — (k — i — 1) > 1,so Wi+1 = 0. Let z be a vertex in Wi+1. By the colour degree condition, z must be incident to at least k edges of distinct colours, and in particular, at least k — i distinct coloured edges not using colours in [i]. By (ci+1), z sends at most k — i — 1 edges to Ti+1. So there exists a vertex u e V(M)\Ti+1 = {xj, yj : 1 < j < i} such that uz is an edge with c(uz) g [i]. Without loss of generality, u = yi and we set zi = z. Clearly (ai) holds.

We now show that (bi) holds for any colour j'. If j' = i, then by (bi+1), there is a rainbow matching Mj'+1 ofsize k — i — 1 on Ti+1 such that c(Mj'+1) n ([i] U {/}) = 0. Set Mj, = MJ+1 U xiyi. So Mj, is a rainbow matching on Ti of size k — i and moreover c(Mj,) n ([i — 1] U {j'}) = 0 as required. If j' = i, then by (bi+1), there is a rainbow matching M'c+1z.) of size k — i — 1 on Ti+1 such that c(Mc+1 )) n ([i] U {c(yizi)}) = 0. Set Mi = M'c+y1Zi) U yizi. Note that Mi is the desired rainbow matching.

A. Lo/Discrete Mathematics 338 (2015) 2119-2124

Let wt be an edge with w e W\ and t e Ti. Since G[W] is empty, t e {zi, zi+1,..., zk—1}. By (ci+1), t e {xi+1, xi+2,..., xk—1}. Suppose that t = xi. By (bi+1), there exists a rainbow matching M+ ) of size k — i — 1 on Ti+1 such that c(M+)) n ([i]U{c (yizi)}) = 0.Let M' be the matching {xjyj : j e [i— 1]}UMlc+JiZ.) U{yizi}. Note that M' is a rainbow matching of size k — 1 vertex-disjoint from the edge wxi. This contradicts the fact that G is a counterexample. Hence we have t e {yi, yi+1,..., yk—1} implying (ci).

Therefore we have found z1,..., zk—1. Let w e W1 = 0. Recall the G[W] = 0, so N(w) c {y1,..., yk—1} by (c1), which implies that dc (w) < d(w) < k — 1, a contradiction. □

Corollary 2.3. Every family G of graphs satisfies y(G) < 3.

For colour sets C and integers i, we now define a (C, i)-adapter below, which will be crucial in the proof of Lemma 2.5. Roughly speaking a (C, i)-adapter is a vertex subset W that contains a rainbow matching M with c (M) = C even after removing a vertex in W.

Given i e N and a set C of colours, a vertex subset W c V(G) is said to be a (C, i)-adapter if there exist (not necessarily edge-disjoint) rainbow matchings M1,..., Mt in G[W] such that c(Mi) = C for all i e [i], and given any w e W, there exists i e [i] such that w $ V(Mi). We write C-adapter for (C, |C | + 1)-adapter. Note that a (C, i)-adapter is also a (C, i')-adapter for all i < ii. The following proposition studies some basic properties of (C, i)-adapters.

Proposition 2.4. Let Gbea graph with an edge-colouring c.

(i) Let C = {c1,..., q} be a set of distinct colours. Let W = {xi, yi, zi ,w : i e [i]} be a vertex set such that c(xiyi) = ci = c(ziw) for all i e [i]. Then W is a C-adapter.

(ii) Let i1,... ,ip e N and let C1,..., Cp be pairwise disjoint colour sets. Suppose that Wj is a (Cj, ij)-adapter for all j e [p] and that W1,..., Wp are pairwise disjoint. Then UJ=1 Wj is a (|Jj=1 Cj, maxje[p]{ij})-adapter.

(iii) Let C be a colour set. Suppose that W is a (C, i)-adapter. Suppose that x, y, z e V(G)\W and w e W such that xy, zw e E(G) and c(xy) = c(zw) e C. Then W U {x, y, z} is a (C U {c(xy)}, i + 1)-adapter.

Proof. To prove (i), we simply set Mi = {xjyj : j e [i]\{i}} U {wzi} for all i e [I] and Mi+1 = {xjyj : j e [I]}.

(ii) Let I = max{lj : j e [p]}. Note that each Wj is a (Cj, £)-adapter. For j e [p], let M\,..., Mi be rainbow matchings in G[Wj] such that c(M}t) = Cj for all i e [I], and given any w e Wj, there exists i e [I] such that w e V(Mj). Set Mt = {Jj=1 Mj. So (ii) holds.

(iii) Let M1,..., Mt be rainbow matchings in G[W ] such that c (Mi) = C for all i e [i], and given any w' e W, there exists i e [i] such that w' e V(Mi). Without loss of generality we have w e V(M1). Now set M' = Mi U {xy} for all i e [i] and Mi+1 = M'U {wz}. Hence, W U {x, y, z} is a (C U {c(xy)}, i + 1)-adapter. □

We prove the following lemma. The main idea of the proof is to consider (C, i)-adapters in G with i maximal.

Lemma 2.5. Letk e N and let 2 < y < 3.Let G be a family of graphs closed under vertex/edge deletion with y(G) < Y .Suppose that G e G with

and that G contains a rainbow matching of size k — 1. Further suppose that for all rainbow matchings M of size k — 1 in G, we have dc (v) > k for all v e V (G)\V (M). Then G contains a rainbow matching of size k.

Proof. We proceed by induction on k. It is trivial for k = 1, so we may assume that k > 2.

Let p e N U {0} and let i1, ...,ip e N with i1 > ... > ip and ^^ i < k — 1. Let P = {W1,..., Wp, U} be a vertex partition of V(G). We say that P has parameters (i1, i2,..., ip) if

(a) there exist p pairwise disjoint colour sets C1,..., Cp such that |Q| = ii for all i e [p];

(b) Wi is a Ci-adapter and = 3ii + 1 for all i e [p];

(c) there exists a rainbow matching MU of size k — 1 —Y7i= 11 in G[U] with c(MU) n Ci = 0 for all i e [p];

(d) U\V(Mu) = 0.

Since G contains a rainbow matching M of size k — 1, such a vertex partition exists (p = 0 and U = V (G) say). We now assume that P is chosen such that the string (i1,..., ip) is lexicographically maximal. (Here, we view (a1, a2,..., ap) as (a1, a2,..., ap, 0,..., 0), e.g. (3, 2, 2) < (4, 1) < (4,1, 1).)

Let C1,..., Cp be the sets of colours guaranteed by (a)-(c). Set W = W1 U • • • U Wp and C = |J p=1 Ci. Let i0 = k — 1 — 1 ii. By (b) and Proposition 2.4(ii), W is a (C, i1 + 1)-adapter. The following claim gives some useful properties of the rainbow matchings in G[U] and G\W. This will be needed to finish the proof of the lemma.

Claim 2.6. (i) Let MU be a rainbow matching of size i0 in G[U ] with c(MU) n C = 0. If |U | > 2i0 + 2 and there is an edge

wz e E(G) with w e W andz e U\V(MU), then we have c(wz) e C. (ii) Let M' be a rainbow matching of size k — 1 — i1 in G\W with c(M') n C1 = 0. If wx e E(G) with w e W1 and x e V(G)\(W1 U V(M')), then c(wx) e C1.

Proof of Claim. Suppose that (i) is false. There exists an edge wz e E(G) such that c(wz) V C, w e W for some i e [p] and z e U\V(My). Note that there exists a rainbow matching MW in G[W\w] such that c(MW) = C since W is a C-adapter. If c(wz) V C U c(MU), then MU U MW U {wz} is a rainbow matching of size k, so we are done. If c(wz) e c(MU), then let xy be the edge in MU such that c(xy) = c(wz). Set W' = Wi U {x, y, z}, Wj = Wj for all j e [p]\{i} and U' = U\{x, y, z}. Let l' = ti + 1 and let l' = ij for all j e [p]\{i}. Set C' = Ci U {c (xy)} and Cj = Cj for all j e [p]\{i}. By Proposition 2.4(iii), Wj is a Cj-adapter for all j e [p]. Note that My = MU — xy is a rainbow matching in G[U'] with c (My ) n Cj = 0 for all j e [p]. Also U'\V(MU/) = U\(V(MU) U {z}) = 0. By relabelling the sets W' and Cj if necessary, we deduce that the vertex partition P' = {W1,..., Wp, U'} has parameters (l[,..., l'p) > (l1,..., lp), which contradicts the maximality of P. Hence (i) holds.

A similar argument proves (ii). □

Suppose that |U | > y(l0 + 1),so |U | > 2l0 + 3.Let H be the resulting subgraph of G[U] obtained after removing all edges of colours in C. Let MU be a rainbow matching in H of size l0 with c(MU) n C = 0, which exists by (c). By Claim 2.6(i), we have for all z e V (H )\V (MU), dcH (z) > k — |C | = l0 + 1. Since y(G) < Y, H contains a rainbow matching M0 of size l0 and a disjoint edge e. If c(e) = c(xy) for some xy e M0, then set Wp+1 = V(e) U {x, y}, Cp+1 = {c(xy)}, and U' = U\(V(e) U {x, y}). Observe that Wp+1 is a Cp+1-adapter by Proposition 2.4(i). Note that M0 — xy is a rainbow matching of size l0 — 1 in G[U'] with c(M0) n Uj-g^] Q = 0 and |U'\V(M0)| = |U| — 2l0 — 2 > 1. Hence the vertex partition P' = {W1,..., Wp+1, U'} has parameters (l1,..., lp, 1), contradicting the maximality of P. If c(e) V c(M0), then M0 U e is a rainbow matching with c(M0 U e) n C = 0. Together with (b), G contains a rainbow matching of size k with colours c(M0 U e) U C, so we are done. Therefore we may assume that

|U|< Y(lo + 1). (1)

Since 2 < y < 3 and l0 < k — 1, by the assumptions of Lemma 2.5, we have |G| > (2 + y/2)k > yk > |U|. Therefore, W = 0 and l1 > 1.

Next, suppose that (y — 2)l1 > 2, so |W1| = 3l1 + 1 < (2 + Y/2)l1.Let H1 be the subgraph of G obtained by removing all vertices of W1 and all edges of colours in C1. By the assumptions of Lemma 2.5, we then have

,2 + YW . ,.2(4 — Y^

IH1I = |G| — |W1| > (2 + 2) (k —11) + (( —2Y)) — 3 + y.

2r " (Y — 2)2

By (b) and (c), H1 contains a rainbow matching M' of size k — 1 — l1. By Claim 2.6(ii), c(wx) e C1 for all w e W1 and x e V(H1)\V(M'). Hence, dcH^ (z) > k — |C1| = k — l1 for all z e V(H1)\V(M'). Note that this statement also holds for any rainbow matchings M' of size k — 1 — l1 in H1. Hence H1 satisfies the hypothesis of the lemma with k = k — l1. By the induction hypothesis, H1 contains a rainbow matching M" of size k —11. By (b), there exists a rainbow matching M1 of size l1 in G[W1] such that c(M1) = C1. Since c(M1) n c(M") c C1 n c(H1) = 0, M1 U M" is a rainbow matching of size k as required. Therefore we may assume that

(Y — 2)l1 < 2. (2)

Recall that W is a (C, l1 + 1)-adapter. So there exist rainbow matchings M*, M*.....Ml*+1 such that c(M*) = C for all

i e [l1 + 1] and

W = U (W\V(M*)). (3)

Let My be a rainbow matching of size l0 in G[U] with c(My) n C = 0 (which exists by (c)). By (d), there exists z e U\V(My). Note that z sends at least dc (z) — |V(My )| > k — 2l0 edges of distinct colours to V(G)\V(My ).Let q = (k — 2l0)/(l1 + 1)1. By (3) and an averaging argument, there exists i e [l1 + 1] such that there exist vertices x1,..., xq e V(G)\V(My U M*) such that c (zxj) is distinct for each j e [q]. By Claim 2.6(i), we have c (zxj) e C = c (M*) for all j e [q]. Let e1,..., eq be edges of M* such that c(ej) = c(zx^ for all j e [q]. Set W' = Qje[q] (V (ej) U {xjS z}) and C' = {c(ej) : j e [q]}. By Proposition 2.4(i), W' is a C'-adapter. Set U' = V(G)\W' and My = (M* U JViU)\W'. Note that V(My) c U' and My is a rainbow matching of size k — 1 — q with c(My') n C' = 0. Therefore, the vertex partition P' = {W', U'} has parameter (q). By the maximality of P, we have l1 > q > (k — 2l0)/(l1 + 1) and so

lo > (k — l1(l1 + 1))/2. (4)

Recall that W = 3lH + 1 < 4lH for all i e [p], that lH +10 = k — 1, and that 2 < y < 3. Finally, we have

p (1) p

|G| = |W1| + |Wi| + |U| < 3l1 + 1 + ^li + Y(lo + 1) i=2 i=2 = 3l1 + 1 + 4(k — 1 — l1) — (4 — Y)lo + Y

(4)4, (4 — Y)(k — l1(l1 + 1)) , < 4k — 3 — l1---h Y

A. Lo/Discrete Mathematics 338 (2015) 2119-2124

Y_ , , (4 — Y)h(h + 1) ^ = (2 + 2)k — 3 —11 +-2-+ Y

(n , Y ^ , (4 — Y)i2 Q , (2) /_ , Y , 2(4 — y) _ ,

< (2 + 2)k + — 3 + Y< (2 + 2)k + V—W — 3 + Y,

a contradiction. This completes the proof of the lemma. □

We are now ready to prove Theorems 1.1 and 1.2.

Proof of Theorems 1.1 and 1.2. We first prove Theorem 1.1 by induction on k. Let G be an edge-coloured graph on n > 7k/2 + 2 vertices with 5c (G) > k. This is trivial for k = 1 and so we may assume that k > 2. By the induction hypothesis G contains a rainbow matching of size k — 1. Since 5c (G) > k, Corollary 2.3 implies that G satisfies the hypothesis of Lemma 2.5 with y = 3. Therefore, G contains a rainbow matching of size k as required.

To prove Theorem 1.2, first note that by Proposition 2.1, y(G') = 2, where G' is the family of all bipartite graphs. Also, for y = 2 + 2s, we have

(2 + 2) k + — 3 + Y = (3 + s)k + — 1 + 2s < (3 + s)k + s—

Therefore, Theorem 1.2 follows from a similar argument used in the preceding paragraph, where we take y = 2 + 2s and G to be the family of all bipartite graphs in the application of Lemma 2.5. □

We would like to point out that an improvement of Corollary 2.3 would lead to an improvement of Theorem 1.1. However, we believe that new ideas are needed to prove the case when 2k < |G| < 3k.

3. Existence of rainbow matching covers

Proof of Theorem 1.3. By colouring every missing edge in G with a new colour, we may assume that G is an edge-coloured complete graph on n vertices with 4mon(G) = t and colours {1, 2,..., p}. For i < p, let Gi be the subgraph of G induced by the edges of colour i. Without loss of generality, we may assume that e(Gi) > e(G2) > • • • > e(Gp).

For 1 < i < p, suppose that we have already found a set M = {M1,..., M'n/2j} of edge-disjoint (possibly empty) rainbow matchings such that U1<j<'tn/2j Mj = Uj'<iEG). We now assign edges of Gi to these matchings so that the resulting rainbow matchings M1,..., M'tn/2j contain all edges of G1 U ... U Gi. Define an auxiliary bipartite graph H as follows. The vertex classes of H are E(Gi) and M. An edge f e E(Gi) is joined to a rainbow matching Mj e M if and only iff is vertex-disjoint from Mj. If H contains a matching of size e(Gi), then we assign f e E(Gi) to Mj e M according to the matching in H. Thus we have obtained the desired rainbow matchings M'v ..., M'tn/2j. Therefore, to prove the theorem, it is sufficient to show that H satisfies Hall's conditions.

Let f e E(Gi). Since f is incident to 2(n — 2) edges in G, f is incident to at most 2(n — 2) matchings Mj e M. Thus,

\Nh(f)l>IM\— 2(n — 2) > (t — 4)n/2. (5)

We divide the proof into two cases depending on the value of i.

Case 1: i < . Let S c E(Gi) with S = 0. Note that each Mj e M has size at most i — 1. If S contains a matching of size 2i — 1, then for every Mj e M,there exists an edge f e S vertex-disjoint from Mj.Thus, NH (S) = M and so |NH (S)| = 'tn/2J > e(Gi) > |S|.

Therefore, we may assume that S does not contain a matching of size 2i — 1. By Vizing's theorem, |S| < 2(i — 1)(A(Gi) + 1) < 2(i — 1)(t + 1). By (5) and the assumption on i, we have

\Nh(S)l > (t — 4)n/2 > 2(i — 1)(t + 1) > \S\.

Therefore, Hall's condition holds for this case.

Case 2: i > .Since e(Gv) > e(G2) >•••> e(GF), we have e(Gi) < (ty/i < 2(t + 1)n/(t — 4). Let S c E(Gl) with S = 0. By (5) and the fact that t > 11, we have

\Nh(S)| > (t — 4)n/2 > 2(t + 1)n/(t — 4) > e(Gi) > \S|.

Therefore, Hall's condition also holds for this case. This completes the proof of the theorem. □

Acknowledgements

The author would like to thank the referees for their helpful suggestions. The research leading to these results was supported by the European Research Council under the ERC Grant Agreement No. 258345.

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