Eulerian and Hamiltonian paths and cycles are key concepts in graph theory. They help us understand how we can traverse graphs by visiting edges or vertices. These ideas have real-world applications in network design, route planning, and even DNA sequencing.
While Eulerian paths focus on visiting every edge once, Hamiltonian paths aim to visit every vertex once. Eulerian problems are generally easier to solve, but Hamiltonian problems are more complex. Understanding both helps us tackle various graph-related challenges in computer science and beyond.
Eulerian vs Hamiltonian Paths and Cycles
Definitions and Key Differences
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visits every edge exactly once in a graph
forms a closed path visiting every edge exactly once
visits every vertex exactly once in a graph
forms a closed path visiting every vertex exactly once
Eulerian concepts focus on edges while Hamiltonian concepts focus on vertices
Eulerian paths/cycles named after Leonhard Euler who solved problem (1736)
Hamiltonian paths/cycles named after William Rowan Hamilton who invented (1857)
Complexity and Applications
Eulerian paths/cycles depend on vertex degrees making them easier to characterize
Hamiltonian paths/cycles depend on complex structural properties making them NP-complete problems
Finding Eulerian paths/cycles generally simpler than Hamiltonian counterparts
Applications include network design (computer networks), optimization problems (delivery routes), and DNA sequencing (genome assembly)
Conditions for Eulerian and Hamiltonian Graphs
Eulerian Path and Cycle Conditions
has Eulerian cycle if all vertices have even degree
Connected graph has Eulerian path if exactly zero or two vertices have odd degree
Two odd-degree vertices serve as start and end points of path
Directed graphs require for Eulerian cycle
Every vertex must have equal in-degree and out-degree
Directed graphs allow Eulerian path under specific degree conditions
At most one vertex has (out-degree) - (in-degree) = 1
At most one vertex has (in-degree) - (out-degree) = 1
All other vertices have equal in-degree and out-degree
Hamiltonian Path and Cycle Conditions
Necessary condition for Hamiltonian cycle requires connected graph with at least 3 vertices
Sufficient conditions include Dirac's theorem and
Dirac's theorem states minimum degree ≥ n/2 (n vertices)
Ore's theorem requires sum of degrees of non-adjacent vertices ≥ n
Graph closure concept introduced by Bondy and Chvátal aids in determining Hamiltonicity
Involves adding edges between non-adjacent vertices with sufficient degree sum
Finding Eulerian and Hamiltonian Paths
Algorithms for Eulerian Paths and Cycles
efficiently finds Eulerian cycles in undirected graphs
Start at any vertex and follow unused edges until returning to start
Recursively traverse any remaining circuits
constructs Eulerian paths/cycles
Choose non-bridge edges whenever possible
Ensures all edges used exactly once
extends Eulerian path finding
Solved using matching algorithms for non-Eulerian graphs
Algorithms for Hamiltonian Paths and Cycles
(DFS) finds Hamiltonian paths/cycles
Worst-case time complexity exponential
solves
Uses dynamic programming
Finds minimum-weight Hamiltonian cycle in O(n^2 * 2^n) time
Approximation algorithms find near-optimal solutions in polynomial time
for Traveling Salesman Problem
Heuristic approaches quickly find good Hamiltonian cycles in large graphs
commonly used in practice
Proofs for Eulerian and Hamiltonian Graphs
Eulerian Graph Proofs
Prove connected graph has Eulerian cycle iff all vertices have even degree
Use and construct cycle
Demonstrate graph has Eulerian path iff exactly zero or two vertices have odd degree
Consider degrees of start and end vertices
Hamiltonian Graph Proofs
Prove Dirac's theorem for Hamiltonian cycles
Show if G has n ≥ 3 vertices and minimum degree δ(G) ≥ n/2, then G Hamiltonian
Establish Ore's theorem for Hamiltonian cycles
Prove if G has n ≥ 3 vertices and d(u) + d(v) ≥ n for non-adjacent vertices u and v, then G Hamiltonian