Tight minimum degree conditions forcing perfect matchings in. matchings in uniform hypergraphs
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1 Tight minimum degree conditions forcing perfect matchings in uniform hypergraphs University of Birmingham 11th September 013 Joint work with Yi Zhao (Georgia State
2 Advertisement: Birmingham Fellowship A Birmingham Fellowship in Combinatorics will be advertised in the near future. These are permanent positions: essentially you are appointed as a Lecturer/Senior Lecturer, but have a light teaching load (and no admin! for the first five years to focus on excellent research. After five years you become a standard Lecturer/Senior Lecturer. If you are interested, please ask me for more information.
3 Characterising graphs with perfect matchings Hall s Theorem characterises all those bipartite graphs with perfect matchings. Tutte s Theorem characterises all those graphs with perfect matchings.
4 Perfect matchings in k-uniform hypergraphs for k 3 decision problem NP-complete (Garey, Johnson 79 Natural to look for simple sufficient conditions
5 minimum l-degree conditions H k-uniform hypergraph, 1 l < k d H (v 1,..., v l = # edges containing v 1,..., v l minimum l-degree δ l (H = minimum over all d H (v 1,..., v l δ 1 (H = minimum vertex degree δ k 1 (H = minimum codegree
6 minimum l-degree conditions H k-uniform hypergraph, 1 l < k d H (v 1,..., v l = # edges containing v 1,..., v l minimum l-degree δ l (H = minimum over all d H (v 1,..., v l δ 1 (H = minimum vertex degree δ k 1 (H = minimum codegree
7 minimum l-degree conditions H k-uniform hypergraph, 1 l < k d H (v 1,..., v l = # edges containing v 1,..., v l minimum l-degree δ l (H = minimum over all d H (v 1,..., v l δ 1 (H = minimum vertex degree δ k 1 (H = minimum codegree δ 1 (H = and δ (H = 1
8 minimum vertex degree results Theorem (Khan and Kühn, Osthus and T. (013 n 0 N s.t if H 3-uniform, n := H n 0 and ( ( n 1 n/3 δ 1 (H > then H contains a perfect matching. Hán, Person and Schacht (009 proved asymptotic version Minimum vertex degree condition tight
9 H n/3 + 1 n/3 1 δ 1 (H = ( n 1 ( n/3 no perfect matching
10 More recent developments Khan (011+ determined the exact minimum vertex degree which forces a perfect matching in a 4-uniform hypergraph. Alon, Frankl, Huang, Rödl, Ruciński, Sudakov (01 gave asymptotically exact threshold for 5-uniform hypergraphs. No other exact vertex degree results are known. (Best known general bounds are due to Kühn, Osthus and Townsend (013+.
11 minimum codegree conditions Theorem (Rödl, Ruciński and Szemerédi (009 H k-uniform hypergraph, H = n sufficiently large, k n δ k 1 (H n/ = perfect matching In fact, they gave exact minimum codegree threshold that forces a perfect matching.
12 Type 1 A A odd edges hit A in even no. of vertices B A B δ k 1 (H H / but no perfect matching
13 Type A blankblank A odd, H /k even or A even, H /k odd edges hit A in odd no. of vertices B A B δ k 1 (H H / but no perfect matching
14 minimum l-degree conditions Theorem (Pikhurko (008 Suppose H k-uniform hypergraph on n vertices and k/ l k 1. ( n l δ l (H (1/ + o(1 = perfect matching k l Previous examples shows result essentially best-possible.
15 minimum l-degree conditions Let δ(n, k, l denote the max. value of δ l (H amongst all k-uniform hypergraphs H on n vertices of Type 1 or. Theorem (T., Zhao (013 Let n be sufficiently large. Suppose H k-uniform hypergraph on n vertices and k/ l k 1. δ l (H > δ(n, k, l = perfect matching Our result makes Pikhurko s exact. Our result implies the theorem of Rödl, Ruciński and Szemerédi.
16 Proof overview We will only consider the case of 4-uniform hypergraphs and minimum -degree. H 4-uniform on n vertices and δ (H > δ(n, 4, Absorbing sets Let 0 < ε γ 1. S an absorbing set if S = γn and H[S] contains a perfect matching H[S Q] has a perfect matching for any set Q s.t. Q εn.
17 Theorem (Markström and Ruciński (011 Suppose H 4-uniform on n vertices ( ( 7 n δ (H 16 + o(1 = H contains matching covering all but n vertices. Our proof is therefore easy if we have an absorbing set: Find absorbing set S in H Find a matching M in H S covering almost all vertices Absorb uncovered vertices using S to obtain perfect matching
18 One can show that there is an absorbing set if: (i xy ( (, (1/ + o(1 n ( tuples ab s.t. N H (xy N H (ab o(1n or (ii o(1n pairs xy ( of large degree, i.e. d H (xy (1/ + o(1 ( n. We can therefore assume (i and (ii fail. We then show that this means H is close to one of the extremal hypergraphs (Type 1 or.
19 G ( ( ab cd (ab(cd E(G abcd E(H
20 (i xy ( ( s.t. (1/ o(1 n ( tuples ab s.t. N H (xy N H (ab o(1n or (ii almost all pairs xy ( are s.t. d H (xy (1/ + o(1 ( n. ( ( xy N(xy
21 (i xy ( ( s.t. (1/ o(1 n ( tuples ab s.t. N H (xy N H (ab o(1n or (ii almost all pairs xy ( are s.t. d H (xy (1/ + o(1 ( n. ( ( xy N(xy (i
22 (i xy ( ( s.t. (1/ o(1 n ( tuples ab s.t. N H (xy N H (ab o(1n or (ii almost all pairs xy ( are s.t. d H (xy (1/ + o(1 ( n. ( ( xy N(xy (i
23 (i xy ( ( s.t. (1/ o(1 n ( tuples ab s.t. N H (xy N H (ab o(1n or (ii almost all pairs xy ( are s.t. d H (xy (1/ + o(1 ( n. ( ( xy N(xy (i
24 (i xy ( ( s.t. (1/ o(1 n ( tuples ab s.t. N H (xy N H (ab o(1n or (ii almost all pairs xy ( are s.t. d H (xy (1/ + o(1 ( n. ( ( xy N(xy (i
25 (i xy ( ( s.t. (1/ o(1 n ( tuples ab s.t. N H (xy N H (ab o(1n or (ii almost all pairs xy ( are s.t. d H (xy (1/ + o(1 ( n. ( ( xy N(xy (i
26 ( ( xy N(xy (i Now use this structure in G to conclude H is close to one of the extremal examples. (Needs quite a bit of work! Then minimum -degree condition forces a perfect matching.
27 Open problems Characterise the minimum vertex degree that forces a perfect matching in a k-uniform hypergraph for k 5. What about minimum l-degree conditions for k-uniform H where 1 < l < k/? (Alon, Frankl, Huang, Rödl, Ruciński, Sudakov have some such results.
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