Eulerian Graphs and Related Topics (eBook)

Front Cover 1
Eulerian Graphs and Related Topics, Part 1, Volume 1 4
Copyright Page 5
Contents 12
Preface 8
CHAPTER I. INTRODUCTION 16
CHAPTER II. THREE PILLARS OF EULERIAN GRAPH THEORY 22
Solution of a Problem Concerning the Geometry of Position 33
On the Possibility of Traversing a Line Complex Without Repetition or Interruption 44
From O. Veblen's "Analysis situs" 47
CHAPTER III. BASIC CONCEPTS AND PRELIMINARY RESULTS 48
III.1. Mixed Graphs and Their Basic Parts 49
III.2. Some Relations Between Graphs and (Mixed) (Di)graphs. Subgraphs 56
III.3. Graphs Derived from a Given Graph 60
III.4. Walks, Trails, Paths, Cycles, Trees Connectivity
III.5. Compatibility, Cyclic Order of K*v and Corresponding Eulerian Trails 87
III.6. Matchings, 1-Factors, 2-Factors, 1-Factorizations, 2-Factorizations, Bipartite Graphs 90
III.7. Surface Embeddings of Graphs Isomorphisms
III.8. Coloring Plane Graphs 107
III.9. Hamiltonian Cycles 109
III.10. Incidence and Adjacency Matrices, Flows and Tensions 116
III.11. Algorithms and Their Complexity 119
III.12. Final Remarks 122
CHAPTER IV. CHARACTERIZATION THEOREMS AND COROLLARIES 124
IV.l. Graphs 124
IV.2. Digraphs 131
IV.3. Mixed Graphs 136
IV.4. Exercises 143
CHAPTER V. EULER REVISITED AND AN OUTLOOK ON SOME GENERALIZATIONS 144
V.l. Trail Decomposition, Path/Cycle Decomposition 144
V.2. Parity Results 146
V.3. Double Tracings 147
V.4. Crossing the Border: Detachments of Graphs 148
V.5. Exercises 150
CHAPTER VI. VARIOUS TYPES OF EULERIAN TRAILS 152
VI.l. Eulerian Trails Avoiding Certain Transitions 152
VI.2. Pairwise Compatible Eulerian Trails 185
VI.3. A-Trails in Plane Graphs 220
VI.4. Exercises 321
CHAPTER VII. TRANSFORMATIONS OF EULERIAN TRAILS 326
VII.l. Transforming Arbitrary Eulerian Trails in Graphs 327
VII.2. Transforming Eulerian Trails of a Special Type 334
VII.3. Transformation of Eulerian Trails in Digraphs 370
VII.4. Final Remarks and Some Open Problems 376
VII.5. Exercises 378
Bibliography 382
Index 398