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๋ฌด์ง๊ฐ ์งํฉ ๋ฌธ์ ์์์ ์์์ํ์ ์กฐํฉ๋ก
ํ์๋
ผ๋ฌธ(๋ฐ์ฌ)--์์ธ๋ํ๊ต ๋ํ์ :์์ฐ๊ณผํ๋ํ ์๋ฆฌ๊ณผํ๋ถ,2019. 8. ๊ตญ์
.๋ฅผ ์ ๊ณต์งํฉ์ด ์๋ ๋ถ๋ถ ์งํฉ๋ค์ ๋ชจ์์ด๋ผ ํ ๋, ์ ๋ฌด์ง๊ฐ ์งํฉ์ด๋ ๊ณต์งํฉ์ด ์๋๋ฉฐ ์ ๊ฐ์ ํํ๋ก ์ฃผ์ด์ง๋ ๊ฒ์ผ๋ก ๋ค์ ์กฐ๊ฑด์ ๋ง์กฑํ๋ ๊ฒ์ ๋งํ๋ค. ์ด๊ณ ์ด๋ฉด ๋ฅผ ๋ง์กฑํ๋ฉฐ ๊ฐ ์ ๋ํด ์ด๋ค. ํนํ ์ธ ๊ฒฝ์ฐ, ์ฆ ๋ชจ๋ ๋ค์ด ํํ๋๋ฉด, ๋ฌด์ง๊ฐ ์งํฉ ๋ฅผ ์ ์์ ๋ฌด์ง๊ฐ ์งํฉ์ด๋ผ๊ณ ํ๋ค.
์ฃผ์ด์ง ์งํฉ๊ณ๊ฐ ํน์ ์กฐ๊ฑด์ ๋ง์กฑํ๋ ๋ฌด์ง๊ฐ ์งํฉ์ ๊ฐ์ง๊ธฐ ์ํ ์ถฉ๋ถ ์กฐ๊ฑด์ ์ฐพ๋ ๋ฌธ์ ๋ ํ์ ๊ฒฐํผ ์ ๋ฆฌ์์ ์์๋์ด ์ต๊ทผ๊น์ง๋ ์กฐํฉ์ํ์์ ๊ฐ์ฅ ๋ํ์ ๋ฌธ์ ์ค ํ๋๋ก ์ฌ๊ฒจ์ ธ์๋ค. ์ด๋ฌํ ๋ฐฉํฅ์ผ๋ก์ ๋ฌธ์ ๋ฅผ ๋ฌด์ง๊ฐ ์งํฉ ๋ฌธ์ ๋ผ๊ณ ๋ถ๋ฅธ๋ค. ๋ณธ ํ์๋
ผ๋ฌธ์์๋ ๋ฌด์ง๊ฐ ์งํฉ ๋ฌธ์ ์ ๊ด๋ จํ์ฌ ์์์ํ์ ํ์ ์ ๋ฆฌ์ ์์์ํ์ ๋ค์ ํฌ๋ฆฌ ์ ๋ฆฌ๋ฅผ ์๊ฐํ๊ณ , (ํ์ดํผ)๊ทธ๋ํ์์์ ๋ฌด์ง๊ฐ ๋ฎ๊ฐ์ ๋ฌด์ง๊ฐ ๋
๋ฆฝ ์งํฉ์ ๊ดํ ๊ฒฐ๊ณผ๋ค์ ๋ค๋ฃจ๊ณ ์ ํ๋ค.Let be a finite family of non-empty subsets on the ground set . A rainbow set of is a non-empty set of the form with such that for every and for each . If , namely if all is represented, then the rainbow set is called a full rainbow set of .
Originated from the celebrated Hall's marriage theorem, it has been one of the most fundamental questions in combinatorics and discrete mathematics to find sufficient conditions on set-systems to guarantee the existence of certain rainbow sets. We call problems in this direction the rainbow set problems. In this dissertation, we give an overview on two topological tools on rainbow set problems, Aharoni and Haxell's topological Hall theorem and Kalai and Meshulam's topological colorful Helly theorem, and present some results on and rainbow independent sets and rainbow covers in (hyper)graphs.Abstract i
1 Introduction 1
1.1 Topological Hall theorem . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Topological colorful Helly theorem . . . . . . . . . . . . . . . . . 3
1.2.1 Collapsibility and Lerayness of simplicial complexes . . . 4
1.2.2 Nerve theorem and topological Helly theorem . . . . . . . 5
1.2.3 Topological colorful Helly theorem . . . . . . . . . . . . 6
1.3 Domination numbers and non-cover complexes of hypergraphs . . 7
1.3.1 Domination numbers of hypergraphs . . . . . . . . . . . . 10
1.3.2 Non-cover complexes of hypergraphs . . . . . . . . . . . . 10
1.4 Rainbow independent sets in graphs . . . . . . . . . . . . . . . . 12
1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Collapsibility of non-cover complexes of graphs 16
2.1 The minimal exclusion sequences . . . . . . . . . . . . . . . . . . 16
2.2 Independent domination numbers and collapsibility numbers of
non-cover complexes of graphs . . . . . . . . . . . . . . . . . . . 21
3 Domination numbers and non-cover complexes of hypergraphs 24
3.1 Proof of Theorem 1.3.4 . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Edge-annihilation . . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Non-cover complexes for hypergraphs . . . . . . . . . . . 27
3.2 Lerayness of non-cover complexes . . . . . . . . . . . . . . . . . 30
3.2.1 Total domination numbers . . . . . . . . . . . . . . . . . 30
3.2.2 Independent domination numbers . . . . . . . . . . . . . 33
3.2.3 Edgewise-domination numbers . . . . . . . . . . . . . . . 34
3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Independent domination numbers of hypergraphs . . . . . 35
3.3.2 Independence complexes of hypergraphs . . . . . . . . . . 36
3.3.3 General position complexes . . . . . . . . . . . . . . . . . 37
3.3.4 Rainbow covers of hypergraphs . . . . . . . . . . . . . . 39
3.3.5 Collapsibility of non-cover complexes of hypergraphs . . . 40
4 Rainbow independent sets 42
4.1 Graphs avoiding certain induced subgraphs . . . . . . . . . . . . 42
4.1.1 Claw-free graphs . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2 -free graphs . . . . . . . . . . . . . . . . . 44
4.1.3 Chordal graphs . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.4 -free graphs and -free graphs . . . . . . . . . . . . . 50
4.2 -colourable graphs . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Graphs with bounded degrees . . . . . . . . . . . . . . . . . . . . 55
4.3.1 The case . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 A topological approach . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . 67
Abstract (in Korean) 69
Acknowledgement (in Korean) 70Docto
Toric algebra of hypergraphs
The edges of any hypergraph parametrize a monomial algebra called the edge
subring of the hypergraph. We study presentation ideals of these edge subrings,
and describe their generators in terms of balanced walks on hypergraphs. Our
results generalize those for the defining ideals of edge subrings of graphs,
which are well-known in the commutative algebra community, and popular in the
algebraic statistics community. One of the motivations for studying toric
ideals of hypergraphs comes from algebraic statistics, where generators of the
toric ideal give a basis for random walks on fibers of the statistical model
specified by the hypergraph. Further, understanding the structure of the
generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in
algebraic statistics and to combinatorial discrepancy. Section 6 (open
problems) has been moderately revise
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