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E-grāmata: Pancyclic and Bipancyclic Graphs

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  • Formāts: PDF+DRM
  • Sērija : SpringerBriefs in Mathematics
  • Izdošanas datums: 18-May-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319319513
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Pancyclic and Bipancyclic GraphsPalielināt
  • Formāts: PDF+DRM
  • Sērija : SpringerBriefs in Mathematics
  • Izdošanas datums: 18-May-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319319513
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Thisbook is focused on pancyclic and bipancyclic graphs and is geared toward researchersand graduate students in graph theory. Readers should be familiar with thebasic concepts of graph theory, the definitions of a graph and of a cycle. Pancyclicgraphs contain cycles of all possible lengths from three up to the number ofvertices in the graph. Bipartite graphs contain only cycles of even lengths, abipancyclic graph is defined to be a bipartite graph with cycles of every evensize from 4 vertices up to the number of vertices in the graph. Cutting edgeresearch and fundamental results on pancyclic and bipartite graphs from a widerange of journal articles and conference proceedings are composed in this bookto create a standalone presentation.Thefollowing questions are highlighted through the book:- What is the smallest possible number of edges in apancyclic graph with v vertices?- When do pancyclic graphs exist with exactly onecycle of every possible length?- What i

s the smallest possible number of edges in abipartite graph with v vertices?- When do bipartite graphs exist with exactly one cycle of every possiblelength?

1.Graphs.- 2. Degrees and Hamiltoneity.- 3. Pancyclicity.- 4. Minimal Pancyclicity.- 5. UniquelyPancyclic Graphs.- 6. Bipancyclic Graphs.- 7. Uniquely Bipancyclic Graphs.- 8. Minimal Bipancyclicity.- References.

Recenzijas

In this book, the authors give a simple survey about the sufficient conditions for a graph to be pancyclic (uniquely bipancyclic). Moreover, the authors give the proofs of some classic results which are useful tools to study and generalize cycle problems. Therefore, this book can help students and researchers alike to find inspiration and ideas on pancyclic and bipancyclic problems. (Junqing Cai, Mathematical Reviews, February, 2017)

1 Graphs
1(8)
1.1 Introduction
1(1)
1.2 Graphs: The Basics
1(2)
1.3 Products
3(1)
1.4 Walks, Paths, and Cycles
4(2)
1.5 Colorings and Cycles
6(3)
2 Degrees and Hamiltoneity
9(12)
2.1 A Theorem of Chvatal
9(1)
2.2 A Theorem of Fan
10(2)
2.3 A Theorem of Bondy and Its Generalization
12(9)
3 Pancyclicity
21(14)
3.1 Introduction
21(1)
3.2 Bounds
22(10)
3.3 Pancyclic Graph Products
32(2)
3.4 Open Problems
34(1)
4 Minimal Pancyclicity
35(14)
4.1 Introduction
35(1)
4.2 Minimal Pancyclic Graphs: Small Orders
36(5)
4.2.1 Fewer Than Two Chords
37(1)
4.2.2 Two Chords
37(1)
4.2.3 Three Chords
38(3)
4.3 Four Chords
41(1)
4.4 Five Chords
42(1)
4.5 More General Bounds for Pancyclics
42(7)
5 Uniquely Pancyclic Graphs
49(20)
5.1 Introduction
49(1)
5.2 Small Cases
49(2)
5.3 Outerplanar UPC Graphs
51(2)
5.4 More General UPC Graphs
53(9)
5.5 Cycle Space of a Graph
62(2)
5.6 Bounds on the Number of Edges in a UPC Graph
64(3)
5.7 Open Problems
67(2)
6 Bipancyclic Graphs
69(12)
6.1 Introduction
69(1)
6.2 Edge Number Conditions
69(2)
6.3 Degree Conditions
71(10)
7 Uniquely Bipancyclic Graphs
81(18)
7.1 Introduction
81(1)
7.2 Graphs with Fewer than Two Chords
82(1)
7.3 Two Chords
82(2)
7.4 Three Chords
84(12)
7.5 More Chords: Computer Searches
96(3)
8 Minimal Bipancyclicity
99(8)
8.1 Introduction
99(1)
8.2 Minimal Bipancyclic Graphs with Excess Less than 2
100(1)
8.3 Excess 2
100(1)
8.4 Excess 3
101(1)
8.5 Excess 4
102(1)
8.6 More General Bounds for Bipancyclics
103(2)
8.7 Bipancyclic Graph Products
105(2)
References 107
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