¶ Eulerian and Hamiltonian Paths Cycles
¶ Competitive programming
When you first hear the names Euler and Hamilton, you probably think of famous mathematicians whose contributions shaped entire fields. But in competitive programming, their names evoke something more specific—two types of graph problems that are as elegant as they are deceptive. Eulerian paths and Hamiltonian paths appear simple at a glance, yet behind that simplicity lies centuries of mathematical thought, patterns that reveal themselves slowly, and problem-solving strategies that blend logic with intuition. This course is about exploring that world in detail, peeling back each layer, and learning how to tackle these path-and-cycle problems with the clarity and confidence of someone who truly understands how graphs behave.
What makes Eulerian and Hamiltonian questions so fascinating is the way they force you to rethink how you navigate a graph. With other topics, you often work with shortest paths, connectivity, trees, or some form of dynamic programming. But here, the challenge is different: instead of thinking about the “best” path or the “optimal” path, you start wondering whether a path that uses every edge exactly once even exists. Or if a route that touches every vertex exactly once can be constructed at all. And when it can, what conditions make it possible? How do we build it efficiently, especially under the time constraints of a contest?
The classical story of Euler wandering through Königsberg, analyzing the bridges, and realizing that the real question wasn’t about geography but about structure, has shaped generations of mathematicians and computer scientists. But beyond the story, the real magic is the simplicity of the condition he found: that a connected graph with either zero or exactly two vertices of odd degree accommodates an Eulerian path, and only a graph with all vertices of even degree can support an Eulerian cycle. Something about this feels almost too perfect—like a rule crafted by nature instead of discovered by a person. And in competitive programming, recognizing such structure instantly transforms a problem that seems complicated into one that yields gracefully.
Hamiltonian paths, on the other hand, are a different beast. They don’t have a tidy parity condition, no neat degree-based test, no elegant equivalent of “count the odd vertices.” Finding a Hamiltonian path is famously hard, not just conceptually but computationally. It’s one of those NP-complete problems that lurks in the background of many contest questions. And yet, competitive programming constantly finds creative ways to incorporate them—through special graph classes where the impossible becomes feasible, through bitmasking strategies, through heuristics, through constructive tricks, or through disguised forms where the Hamiltonian nature only becomes apparent when you rearrange the problem in just the right way.
If Euler is the approachable genius who hands you a beautiful pattern, Hamilton is the mysterious thinker who forces you to wrestle with complexity. And grappling with both perspectives is what makes this journey so rewarding.
As you progress through these articles, you’ll notice how often contests rely on Eulerian insights, even when the problem doesn’t explicitly mention edges or cycles. For instance, problems about reconstructing a string from overlapping substrings often hide an Eulerian trail in a de Bruijn graph. Problems about verifying whether a multiset of dominoes can form a chain turn out to be classic Eulerian path questions. Problems about ordering structures, tracing routes through tasks, or balancing in-degree and out-degree in directed graphs are, at their core, Eulerian puzzles. Once you start seeing them, it becomes impossible to unsee them. They appear everywhere—in simple beginner problems as well as in championship-level challenges.
Hamiltonian problems, on the other hand, tend to hide in places where you wouldn’t expect them. Sometimes they appear as “traveling salesman” variants. Sometimes they appear in grid navigation problems, where you must visit each cell exactly once without revisiting. Sometimes they show up in word-arranging or sequence-building tasks where each piece must be used once. Sometimes the Hamiltonian nature is disguised behind constraints that allow dynamic programming over subsets. And sometimes the problem is designed so cleverly that a Hamiltonian structure reveals itself only after reframing the entire challenge.
What makes both kinds of paths so compelling is not the abstract theory behind them but the way they force you to think. They require you to observe patterns that aren’t immediately visible, to detect balance in a graph that looks unbalanced, to trust certain graph invariants even when intuition falters. Competitive programming thrives on such moments—when you recognize a hidden structure that suddenly simplifies the entire problem.
One of the most valuable skills you’ll develop while studying this topic is the ability to reduce unfamiliar problems to known patterns. For Eulerian questions, this often means translating the narrative into a graph where you can track in-degrees and out-degrees, verify connectivity, and check parity. For Hamiltonian problems, it often means cleverly limiting the search space or discovering that the graph belongs to a class where Hamiltonian paths are achievable without brute force—like tournaments, grids with special shapes, or graphs with monotonicity constraints.
You’ll also become more comfortable with constructive strategies. Many Eulerian problems expect you not only to identify that a path exists but to actually build it, often through an implementation of Hierholzer’s algorithm. Building Hamiltonian paths, too, requires careful reasoning—sometimes greedy orderings, sometimes clever pruning, sometimes bitmask DP that feels almost magical when everything aligns.
It’s not just algorithms that matter here. Thinking deeply about these problems strengthens your understanding of graph theory as a whole. You start seeing graphs not just as sets of nodes and edges but as living structures with personality—some balanced, some chaotic, some symmetrical, some stubbornly irregular. You start noticing how slight changes to a graph’s degree distribution radically change the landscape of what’s possible. You start appreciating why certain questions belong to polynomial time while others explode exponentially. And this awareness spills over into all areas of competitive programming. Pattern recognition sharpens. Intuition grows stronger.
One of the delightful things about Eulerian and Hamiltonian problems is how naturally they blend elegance with challenge. Some Eulerian tasks feel like you’re assembling a jigsaw puzzle where every piece fits beautifully once you find the right angle. Others feel more like delicate surgery, where one misstep in the ordering can ruin the entire path. Hamiltonian tasks, meanwhile, often give you the feeling of navigating a maze where every decision must be made with care, because the wrong turn leads to an explosion of possibilities.
The more you grapple with these problems, the more you start appreciating the relationship between constraints and feasibility. A problem that looks impossible on an arbitrary graph becomes solvable when you realize that the graph is actually a tournament or a DAG or a bipartite structure with carefully controlled degrees. Much of competitive programming success comes from learning to spot these hidden constraints quickly.
A full journey through Eulerian and Hamiltonian paths isn’t complete without wrestling with directed graphs. Directed versions of these problems can be surprisingly tricky. With Eulerian paths, the balance between in-degree and out-degree introduces new insights and new pitfalls. With Hamiltonian paths, direction adds layers of complexity that often require clever dynamic programming. But mastering directed graphs will push your understanding to a deeper level, preparing you for some of the most sophisticated contest problems involving ordering, scheduling, and path construction.
One of the themes that will recur throughout this course is the difference between existence and construction. Sometimes the challenge lies in simply determining whether a path exists at all. Other times, the real task is constructing the actual path, often under tight time constraints or with limitations on memory. Good competitive programmers learn to handle both sides—checking existence efficiently and building solutions methodically.
You will also come across the subtle beauty of cycles. A cycle isn’t just a path that loops back to its start—it’s a statement of balance, structure, and symmetry. Eulerian cycles feel almost poetic in their simplicity once the degree conditions are met. Hamiltonian cycles, by contrast, feel like puzzles that reward persistence and creativity. Understanding how cycles behave deepens your intuition about how graphs can be assembled, traversed, or decomposed.
As you go deeper, you may find yourself developing an appreciation for the historical roots of these problems. Euler's work laid the foundation of graph theory itself, while Hamilton’s puzzle-like constructions pushed mathematicians to think differently about paths that touch every node. Knowing the history isn’t necessary for solving problems, but it does add a sense of connection to a long tradition of mathematical exploration.
One of the most rewarding aspects of studying these paths is watching your thinking evolve. At first, you might rely heavily on checking conditions mechanically. Then you’ll start predicting outcomes instinctively. Eventually, you’ll reach a point where you can look at a problem and know—almost immediately—whether it contains an Eulerian trick or a Hamiltonian twist. This transformation is subtle but powerful, and it’s one of the skills that separates seasoned competitors from those still finding their footing.
This 100-article journey is designed not just to teach you formulas or algorithms, but to shape the way you see graphs. It aims to give you a sense of confidence when encountering problems that others find intimidating. You’ll learn to identify structural cues that reveal the nature of the underlying path. You’ll practice converting real-world constraints into graph language. You’ll explore multiple ways to build the same path, understanding when to favor efficiency, simplicity, or stability.
By the time you finish this course, Eulerian and Hamiltonian problems will no longer feel like isolated topics but like familiar companions in your broader problem-solving journey. You’ll recognize when a graph’s degree pattern whispers of an Eulerian cycle. You’ll sense when a clever ordering hints at a hidden Hamiltonian route. And you’ll be comfortable constructing paths and cycles with precision, even under pressure.
More importantly, you’ll be able to approach graph problems with a level of calmness and insight that comes only from deep familiarity. What seems overwhelming today will feel natural by the end. And that’s the true goal—not just technical mastery, but a kind of fluency that lets you navigate graph challenges with ease.
This course invites you into that mindset. A mindset where graphs are not intimidating but intriguing, where path problems are not obstacles but opportunities, and where every challenge is a chance to uncover structure that was always there, waiting to be seen.
Let’s begin the journey.
¶ Eulerian and Hamiltonian Paths-Cycles
¶ Beginner Level:
1. Introduction to Graph Theory in Competitive Programming
2. What is an Eulerian Path and Cycle?
3. Understanding Hamiltonian Path and Cycle
4. Basic Terminology in Graph Theory: Vertices, Edges, and Degree
5. Graphs: Directed vs. Undirected
6. Graph Representation in Competitive Programming
7. The Concept of Degree in Graphs
8. What Makes a Graph Eulerian?
9. What Makes a Graph Hamiltonian?
10. Understanding Path vs. Cycle in Graph Theory
11. The Eulerian Path Problem
12. Conditions for an Eulerian Path in an Undirected Graph
13. Conditions for an Eulerian Cycle in an Undirected Graph
14. Eulerian Paths in Directed Graphs
15. Conditions for an Eulerian Path in a Directed Graph
16. Introduction to the Hierholzer's Algorithm for Eulerian Paths
17. Basic Hamiltonian Path Problem
18. Backtracking for Hamiltonian Path Problems
19. Graph Traversal Techniques for Eulerian and Hamiltonian Problems
20. The Brute Force Approach to Eulerian and Hamiltonian Cycles
¶ Intermediate Level:
21. The Chinese Postman Problem and Eulerian Circuits
22. Eulerian Path: Practical Algorithm
23. Efficiently Finding Eulerian Cycles
24. Depth First Search (DFS) in Graphs
25. Backtracking for Finding Eulerian Cycles
26. Backtracking for Finding Hamiltonian Cycles
27. Necessary and Sufficient Conditions for Hamiltonian Cycles
28. Dynamic Programming Approach to Hamiltonian Path
29. Hamiltonian Path in Directed Graphs
30. Eulerian Path vs. Hamiltonian Path: Key Differences
31. Using DFS for Hamiltonian Path Verification
32. Hamiltonian Cycle Problem in Complete Graphs
33. The Role of Degrees in Eulerian and Hamiltonian Paths
34. Finding Eulerian Path in Directed Multigraphs
35. The Use of Hierholzer's Algorithm in Solving Eulerian Path Problems
36. Using the Fleury’s Algorithm for Eulerian Circuits
37. Hamiltonian Path in Bipartite Graphs
38. The Relationship Between Eulerian and Hamiltonian Cycles
39. Properties of Eulerian Cycles in Planar Graphs
40. Hamiltonian Cycle in Sparse Graphs
¶ Advanced Level:
41. Advanced Backtracking for Finding Eulerian and Hamiltonian Cycles
42. Approximation Algorithms for Hamiltonian Path Problems
43. Greedy Strategies for Eulerian Paths and Cycles
44. Hamiltonian Cycle Problem in NP-Complete Graphs
45. Using Dynamic Programming with Bitmasking for Hamiltonian Path
46. Efficient Algorithms for Eulerian Paths in Large Graphs
47. Eulerian Path in Weighted Graphs
48. The Use of Eulerian Circuits in Network Design
49. Hamiltonian Path in Graphs with Constraints
50. Graph Coloring and Its Relationship to Eulerian Paths
51. Applications of Eulerian and Hamiltonian Cycles in Routing Problems
52. Complexity of Hamiltonian Cycle in Graph Theory
53. A Search Algorithm for Finding Hamiltonian Path*
54. Eulerian Path and Cycle in Random Graphs
55. The NP-Hardness of the Hamiltonian Path Problem
56. Advanced Dynamic Programming for Eulerian Cycles
57. Exploring Hamiltonian Path in 3D Graphs
58. Eulerian Path Algorithms in Real-Time Systems
59. Advanced Backtracking: Solving Large Hamiltonian Cycle Problems
60. Parameterized Complexity of Eulerian and Hamiltonian Cycles
61. Hamiltonian Path in Dense Graphs
62. Handling Edge Cases in Eulerian and Hamiltonian Problems
63. Eulerian and Hamiltonian Cycles in Dense Networks
64. Applications of Eulerian and Hamiltonian Cycles in Robotics
65. Computational Complexity of Finding Eulerian Cycles
66. Parallel Algorithms for Hamiltonian Path Problems
67. Using Depth-First Search for Hamiltonian Path Verification
68. Eulerian Circuits and Their Role in Circuit Design
69. Hamiltonian Cycles in Grid Graphs
70. Using Kirchhoff's Matrix-Tree Theorem for Eulerian Cycles
71. Hamiltonian Cycle Approximation in Planar Graphs
72. Combining DFS and BFS for Efficient Hamiltonian Path Algorithms
73. Eulerian Path and Cycle in Graph Coloring Problems
74. Hamiltonian Cycle in Weighted Graphs
75. The Role of Minimum Spanning Trees in Eulerian and Hamiltonian Paths
76. Graph Decomposition for Eulerian Cycles
77. Stochastic Algorithms for Finding Hamiltonian Paths
78. Quantum Algorithms for Eulerian Path Problems
79. Polynomial Time Approximation Schemes for Hamiltonian Path
80. Eulerian Cycle with Dynamic Graph Changes
81. Advanced Approaches to Hamiltonian Cycle in Randomized Graphs
82. Greedy Approaches to Eulerian Cycles in Networks
83. Handling Multiple Solutions in Hamiltonian Path Problems
84. Hamiltonian Path and Cycles in High-Dimensional Graphs
85. Iterative Deepening for Finding Eulerian Paths
86. Using Cut-Vertices to Find Hamiltonian Paths
87. Application of Eulerian and Hamiltonian Cycles in Game Theory
88. Quantum Computing Approaches to Hamiltonian Path Problems
89. Hamiltonian Path and Cycles in Directed Acyclic Graphs
90. Solving Hamiltonian Path Problems Using Local Search Algorithms
91. Computational Geometry and Eulerian Cycles
92. Hamiltonian Cycle with Constraints: Solving with Integer Programming
93. Optimal Approximation for Eulerian Paths in Networks
94. Hamiltonian Path in 3D Graphs Using Spatial Partitioning
95. Combinatorial Algorithms for Finding Eulerian Cycles
96. The Use of Eulerian Circuits in Traveling Salesman Problems
97. Advanced Hamiltonian Cycle Algorithms Using Labeled Graphs
98. Handling Dense Graphs in Hamiltonian Path Problems
99. Approximation Algorithms for Hamiltonian Cycle in Large Graphs
100. Future Trends in Eulerian and Hamiltonian Path Algorithms