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Holeμ κ΄μ μμ κ·Έλνμ μ ν₯κ·Έλνμ ꡬ쑰μ κ΄ν μ°κ΅¬
νμλ
Όλ¬Έ(λ°μ¬)--μμΈλνκ΅ λνμ :μ¬λ²λν μνκ΅μ‘κ³Ό,2019. 8. κΉμλ Ή.μ΄ λ
Όλ¬Έμμλ μ ν₯κ·Έλνμ κ·Έλνμ νμ κ΄μ μμ κ³ν΅λ°μ κ·Έλνμ κ·Έλνμ μΌκ°νμ λνμ¬ μ°κ΅¬νλ€.
κΈΈμ΄ 4 μ΄μμΈ μ λλ μΈμ΄ν΄μ νμ΄λΌ νκ³ νμ΄ μλ κ·Έλνλ₯Ό μΌκ°νλ κ·ΈλνλΌ νλ€. ꡬ체μ μΌλ‘, μΈμ΄ν΄μ κ°μ§ μλ μ ν₯κ·Έλνμ κ³ν΅λ°μ κ·Έλνκ° μΌκ°νλ κ·ΈλνμΈμ§ νμ νκ³ , μ£Όμ΄μ§ κ·Έλνλ₯Ό μΌκ°ννμ¬ ν΄λ¦μκ° ν¬κ² μ°¨μ΄ λμ§ μλ κ·Έλνλ₯Ό λ§λλ λ°©λ²μ μ°Ύκ³ μ νλ€. μ΄ λ
Όλ¬Έμ μ°κ΅¬ λ΄μ©μ λ°λΌ λ λΆλΆμΌλ‘ λλλ€.
λ¨Όμ μ ν₯κ·Έλνμ μ ν₯κ·Έλνμ κ³ν΅λ°μ κ·Έλνλ₯Ό μμ νκ² νΉμ§ννκ³ , μ ν₯κ·Έλν μ λͺ¨λ μ ν₯λ³μμ λ°©ν₯μ μ κ±°ν κ·Έλνκ° μΌκ°νλ κ·Έλνμ΄λ©΄, μ κ³ν΅λ°μ κ·Έλν μμ μΌκ°νλ κ·Έλνμμ 보μλ€. λν μ μ μμ μΌκ°νμ κ°λ μ°κ²°λ κ·Έλνμ κ³ν΅λ°μμλ₯Ό κ³μ°ν μ 리λ₯Ό νμ₯νμ¬ λ§μ μμ μΌκ°νμ ν¬ν¨ν μ°κ²°λ κ·Έλνμ κ³ν΅λ°μμλ₯Ό κ³μ°νμλ€.
λ€λ₯Έ ν νΈ κ·Έλν μ λΉμΌκ°ν μ§μ μ λνμ¬ λ₯Ό λ§μ‘±νλ μ μΌκ°νλ κ·Έλν κ° μ‘΄μ¬ν¨μ 보μλ€.
κ·Έλ¦¬κ³ μ΄λ₯Ό λκ΅¬λ‘ μ΄μ©νμ¬ NC propertyλ₯Ό λ§μ‘±νλ κ·Έλνκ° Hadwiger μΆμΈ‘κ³Ό Erd\H{o}s-Faber-Lov\'{a}sz μΆμΈ‘μ λ§μ‘±ν¨μ μ¦λͺ
νκ³ , λΉμΌκ°ν μ§μκ° μ κ³μΈ κ·Έλνλ€μ΄ linearly -boundedμμ μ¦λͺ
νμλ€.This thesis aims at studying phylogeny graphs and graph completions in the aspect of holes of graphs or digraphs. A hole of a graph is an induced cycle of length at least four and a graph is chordal if it does not contain a hole. Specifically, we determine whether the phylogeny graphs of acyclic digraphs are chordal or not and find a way of chordalizing a graph without increasing the size of maximum clique not so much. In this vein, the thesis is divided into two parts.
In the first part, we completely characterize phylogeny graphs of digraphs and digraphs, respectively, for a positive integer . Then, we show that the phylogeny graph of a digraph is chordal if the underlying graph of is chordal for any positive integer . In addition, we extend the existing theorems computing phylogeny numbers of connected graph with a small number of triangles to results computing phylogeny numbers of connected graphs with many triangles.
In the second part, we present a minimal chordal supergraph of a graph satisfying the inequality for the non-chordality index of . Using the above chordal supergraph as a tool, we prove that the family of graphs satisfying the NC property satisfies the Hadwiger conjecture and the Erd\H{o}s-Faber-Lov\'{a}sz Conjecture, and the family of graphs with bounded non-chordality indices is linearly -bounded.Contents
Abstract i
1 Introduction 1
1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Phylogeny graphs . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Graph colorings and chordal completions . . . . . . . . 14
2 Phylogeny graphs 19
2.1 Chordal phylogeny graphs . . . . . . . . . . . . . . . . . . . . 19
2.1.1 (1,j) phylogeny graphs and (i,1) phylogeny graphs . . 20
2.1.2 (2,j) phylogeny graphs . . . . . . . . . . . . . . . . . . 28
2.2 The phylogeny number and the triangles and the diamonds of
a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 A new minimal chordal completion 61
3.1 Graphs with the NC property . . . . . . . . . . . . . . . . . . 64
3.2 The ErdΛ os-Faber-LovΓ‘sz Conjecture . . . . . . . . . . . . . . . 73
3.3 A minimal chordal completion of a graph . . . . . . . . . . . . 80
3.3.1 Non-chordality indices of graphs . . . . . . . . . . . . . 80
3.3.2 Making a local chordalization really local . . . . . . . . 89
3.4 New Ο-bounded classes . . . . . . . . . . . . . . . . . . . . . . 97
Abstract (in Korean) 107Docto
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