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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
Further results on packing related parameters in graphs
Given a graph G = (V, E), a set B subset of V (G) is a packing in G if the closed neighborhoods of every pair of distinct vertices in B are pairwise disjoint. The packing number rho(G) of G is the maximum cardinality of a packing in G. Similarly, open packing sets and open packing number are defined for a graph G by using open neighborhoods instead of closed ones. We give several results concerning the (open) packing number of graphs in this paper. For instance, several bounds on these packing parameters along with some Nordhaus-Gaddum inequalities are given. We characterize all graphs with equal packing and independence numbers and give the characterization of all graphs for which the packing number is equal to the independence number minus one. In addition, due to the close connection between the open packing and total domination numbers, we prove a new upper bound on the total domination number gamma(t)(T) for a tree T of order n >= 2 improving the upper bound gamma(t)(T) <= (n + s)/2 given by Chellali and Haynes in 2004, in which s is the number of support vertices of T
Structure and properties of maximal outerplanar graphs.
Outerplanar graphs are planar graphs that have a plane embedding in which each vertex lies on the boundary of the exterior region. An outerplanar graph is maximal outerplanar if the graph obtained by adding an edge is not outerplanar. Maximal outerplanar graphs are also known as triangulations of polygons. The spine of a maximal outerplanar graph G is the dual graph of G without the vertex that corresponds to the exterior region. In this thesis we study metric properties involving geodesic intervals, geodetic sets, Steiner sets, different concepts of boundary, and also relationships between the independence numbers and domination numbers of maximal outerplanar graphs and their spines. In Chapter 2 we find an extension of a result by Beyer, et al. [3] that deals with Hamiltonian degree sequences in maximal outerplanar graphs. In Chapters 3 and 4 we give sharp bounds relating the independence number and domination number, respectively, of a maximal outerplanar graph to those of its spine. In Chapter 5 we discuss the boundary, contour, eccentricity, periphery, and extreme set of a graph. We give a characterization of the boundary of maximal outerplanar graphs that involves the degrees of vertices. We find properties that characterize the contour of a maximal outerplanar graph. The other main result of this chapter gives characterizations of graphs induced by the contour and by the periphery of a maximal outerplanar graph. In Chapter 6 we show that the generalized intervals in a maximal outerplanar graph are convex. We use this result to characterize geodetic sets in maximal outerplanar graphs. We show that every Steiner set in a maximal outerplanar graph is a geodetic set and also show some differences between these types of sets. We present sharp bounds for geodetic numbers and Steiner numbers of maximal outerplanar graphs
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
A characterization of trees with equal 2-domination and 2-independence numbers
A set of vertices in a graph is a -dominating set if every vertex
of not in is adjacent to at least two vertices in , and is a
-independent set if every vertex in is adjacent to at most one vertex of
. The -domination number is the minimum cardinality of a
-dominating set in , and the -independence number is the
maximum cardinality of a -independent set in . Chellali and Meddah [{\it
Trees with equal -domination and -independence numbers,} Discussiones
Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive
characterization of trees with equal -domination and -independence
numbers. Their characterization is in terms of global properties of a tree, and
involves properties of minimum -dominating and maximum -independent sets
in the tree at each stage of the construction. We provide a constructive
characterization that relies only on local properties of the tree at each stage
of the construction.Comment: 17 pages, 4 figure
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