Graphs & Digraphs (7th Edition) – Complete Solution Manual by Chartrand | Verified Answers 2025/2026

Covers All 21 Chapters




SOLUTIONS MANUAL

, Table of Contents
Preface to the Sixth Edition xi

1. Introduction 1

1.1 Graphs 1
1.2 The Degree of a Vertex 5
1.3 Isomorphic Graphs 7
1.4 Regular Graphs 12
1.5 Bipartite Graphs 13
1.6 Operations on Graphs 16
1.7 Degree Sequences 18
1.8 Multigraphs 25
• Exercises for Chapter 1 28

2. Connected Graphs and Distance 37

2.1 Connected Graphs 37
2.2 Distance in Graphs 44
• Exercises for Chapter 2 51
3. Trees 57

3.1 Nonseparable Graphs 57
3.2 Introduction to Trees 62
3.3 Spanning Trees 69
3.4 The Minimum Spanning Tree Problem 81
• Exercises for Chapter 3 86
4. Connectivity 95

4.1 Connectivity and Edge-Connectivity 95
4.2 Theorems of Menger and Whitney 102
• Exercises for Chapter 4 110
5. Eulerian Graphs 115
5.1 The Königsberg Bridge Problem 115
5.2 Eulerian Circuits and Trails 117
• Exercises for Chapter 5 123
6. Hamiltonian Graphs 125

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, 6.1 Hamilton’s Icosian Game 125
6.2 Sufficient Conditions for Hamiltonian Graphs 128
6.3 Toughness of Graphs 134
6.4 Highly Hamiltonian Graphs 140
6.5 Powers of Graphs and Line Graphs 145
• Exercises for Chapter 6 154
7. Digraphs 161
7.1 Introduction to Digraphs 161
7.2 Strong Digraphs 166
7.3 Eulerian and Hamiltonian Digraphs 167
7.4 Tournaments 169
7.5 Kings in Tournaments 179
7.6 Hamiltonian Tournaments 180
• Exercises for Chapter 7 184
8. Flows in Networks 191

8.1 Networks 191
8.2 The Max-Flow Min-Cut Theorem 199
8.3 Menger Theorems for Digraphs 207
• Exercises for Chapter 8 212
9. Automorphisms and Reconstruction 217

9.1 The Automorphism Group of a Graph 217
9.2 Cayley Color Graphs 223
9.3 The Reconstruction Problem 228
• Exercises for Chapter 9 235
10. Planar Graphs 239
10.1 The Euler Identity 239
10.2 Maximal Planar Graphs 248
10.3 Characterizations of Planar Graphs 252
10.4 Hamiltonian Planar Graphs 264
• Exercises for Chapter 10 268
11. Nonplanar Graphs 275
11.1 The Crossing Number of a Graph 275

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, 11.2 The Genus of a Graph 286
11.3 The Graph Minor Theorem 300
• Exercises for Chapter 11 302
12. Matchings, Independence and Domination 305

12.1 Matchings 305
12.2 1-Factors 310
12.3 Independence and Covers 317
12.4 Domination 322
• Exercises for Chapter 12 329
13. Factorization and Decomposition 335
13.1 Factorization 335
13.2 Decomposition 343
13.3 Cycle Decomposition 345
13.4 Graceful Graphs 351
• Exercises for Chapter 13 358
14. Vertex Colorings 363
14.1 The Chromatic Number of a Graph 363
14.2 Color-Critical Graphs 371
14.3 Bounds for the Chromatic Number 374
• Exercises for Chapter 14 385
15. Perfect Graphs and List Colorings 393
15.1 Perfect Graphs 393
15.2 The Perfect and Strong Perfect Graph Theorems 402
15.3 List Colorings 405
• Exercises for Chapter 15 410
16. Map Colorings 415
16.1 The Four Color Problem 415
16.2 Colorings of Planar Graphs 426
16.3 List Colorings of Planar Graphs 428
16.4 The Conjectures of Ha jós and Hadwiger 434
16.5 Chromatic Polynomials 438
16.6 The Heawood Map-Coloring Problem 444

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