Euler and Hamiltonian paths are key concepts in graph theory. They help us understand how to traverse graphs efficiently, visiting edges or vertices exactly once. These ideas have real-world applications in route planning, puzzle solving, and network design.

Euler paths focus on edges, while Hamiltonian paths deal with vertices. Both have specific conditions for existence and algorithms for finding them. Understanding these paths is crucial for solving complex problems in various fields, from logistics to computer science.

Euler Paths and Circuits

Fundamental Concepts of Euler Paths and Circuits

• Euler path traverses every edge in a graph exactly once
• Euler circuit forms a closed loop traversing every edge exactly once
• Eulerian graph contains an Euler circuit
• Semi-Eulerian graph contains an Euler path but not an Euler circuit

Conditions for Eulerian Graphs

• Necessary condition for Euler path requires all vertices except at most two have even degree
• Sufficient condition for Euler path ensures graph is connected with at most two odd-degree vertices
• Necessary and sufficient condition for Euler circuit mandates all vertices have even degree and graph is connected
• Fleury's algorithm finds Euler circuits in Eulerian graphs

Practical Applications of Euler Paths

• Snow plow route optimization utilizes Euler paths to clear streets efficiently
• Mail delivery routes leverage Euler paths to minimize repeated travel
• Network design employs Euler circuits for efficient data transmission (token ring networks)
• Puzzle solving applies Euler paths in games like "draw without lifting your pencil"

Hamiltonian Paths and Cycles

Core Concepts of Hamiltonian Paths and Cycles

• Hamiltonian path visits every vertex in a graph exactly once
• Hamiltonian cycle forms a closed loop visiting every vertex exactly once
• Hamiltonian graph contains a Hamiltonian cycle
• Semi-Hamiltonian graph contains a Hamiltonian path but not a Hamiltonian cycle

Conditions for Hamiltonian Graphs

• Necessary conditions for Hamiltonian graphs include connectivity and absence of articulation points
• Dirac's theorem provides sufficient condition: graph with n ≥ 3 vertices and minimum degree ≥ n/2 is Hamiltonian
• Ore's theorem offers another sufficient condition: sum of degrees of non-adjacent vertices ≥ n for graph with n ≥ 3 vertices
• Bondy-Chvátal theorem generalizes sufficient conditions for Hamiltonian graphs

Complexity and Algorithms for Hamiltonian Problems

• Determining existence of Hamiltonian paths or cycles belongs to NP-complete class of problems
• Held-Karp algorithm solves Hamiltonian cycle problem in $O(n^2 2^n)$ time
• Nearest neighbor algorithm provides heuristic approach for finding approximate Hamiltonian cycles
• Genetic algorithms offer alternative method for tackling Hamiltonian problems in large graphs

Applications

Optimization Problems in Graph Theory

• Traveling Salesman Problem seeks shortest Hamiltonian cycle in weighted graph
• Nearest Neighbor and 2-opt heuristics provide approximate solutions for Traveling Salesman Problem
• Branch and bound algorithm offers exact solution for smaller instances of Traveling Salesman Problem
• Vehicle routing problems extend Traveling Salesman Problem to multiple vehicles and constraints

Historical and Practical Graph Traversal Problems

• Konigsberg Bridge Problem led to development of Euler path concept
• Seven Bridges of Konigsberg represented as multigraph with land masses as vertices and bridges as edges
• Euler proved impossibility of traversing all bridges exactly once, establishing foundations of graph theory
• Modern applications include optimizing routes for mail delivery and waste collection

Graph Theory in Scientific and Engineering Applications

• Utility graph traversal problem involves connecting houses to utilities without crossing lines
• K3,3 graph represents utility problem, proving its impossibility on a plane
• Molecular structure analysis uses graph theory to represent chemical bonds and atomic arrangements
• Graph isomorphism helps identify structurally equivalent molecules in chemistry