Polytopal graphs and arrangements of curves.
Item
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Title
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Polytopal graphs and arrangements of curves.
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Identifier
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AAI9224826
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identifier
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9224826
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Creator
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Jeong, Dalyoung.
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Contributor
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Adviser: Joseph Malkevitch
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Date
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1992
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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Let P be a 4-valent 3-polytope, and let G be a plane 4-valent 3-connected graph whose vertices and edges correspond to the vertices and edges of P, respectively. Then, {dollar}{dollar}\rm p\sb3 = 8 + \sum\sb{lcub}\rm k \geq 4{rcub}\ (k - 4) \cdot p\sb{lcub}k{rcub},\eqno(*){dollar}{dollar}where p{dollar}\sb{lcub}\rm k{rcub}{dollar} is the number of faces of G (or P) with k sides. The following "Eberhard-type" theorem, which extends a theorem of B. Grunbaum, is proven:;Theorem. Given a collection of non-negative integers p{dollar}\sb3\sp*{dollar}, p{dollar}\sb5\sp*{dollar}, p{dollar}\sb6\sp*{dollar}, {dollar}\...{dollar} p{dollar}\sb{lcub}\rm n{rcub}\sp*{dollar} which satisfies {dollar}(*),{dollar} there exists a non-negative integer p{dollar}\sb4\sp*{dollar} and a 4-valent 3-polytopal graph G, having an Eulerian circuit which is generated by choosing the "middle edge" to be the next edge as one approaches a vertex along an edge, and such that p{dollar}\sb{lcub}\rm k{rcub}{dollar}(G) = p{dollar}\sb{lcub}\rm k{rcub}\sp*{dollar} (3 {dollar}\leq{dollar} k {dollar}\leq{dollar} n). (p{dollar}\sb{lcub}\rm k{rcub}{dollar}(G) denotes the number of faces with k sides in G.).;Similar results for 4-valent 3-polytopal graphs having either a Hamiltonian circuit or a spanning tree without 2-valent vertices are also proven.;Additional structure theorems about plane 4-valent graphs which can be written as the union of simple plane circuit are also proved. For example, the following theorem is proven in detail:;Theorem. If p{dollar}\sb{lcub}\rm s{rcub}\sp*{dollar} = 1, p{dollar}\sb3\sp*{dollar} = 8 + (s {dollar}-{dollar} 4), and p{dollar}\sb{lcub}\rm i{rcub}\sp*{dollar} = 0 for all i {dollar}\not={dollar} 3, 4, and k, then there is no plane 4-valent 3-connected graph G which is the graph of an arrangement of simple curves for which p{dollar}\sb{lcub}\rm k{rcub}{dollar}(G) = p{dollar}\sb{lcub}\rm k{rcub}\sp*{dollar} for all k = 3, 4, 5, ..., s, and any choice of p{dollar}\sb4\sp*{dollar}.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
