Combinatorics & Graph Theory

Combinatorics and graph theory simulations visualise the structure of discrete mathematical objects and the algorithms that operate on them. Graph-colouring simulations apply greedy and backtracking algorithms to assign colours to graph vertices with no two adjacent vertices sharing a shade, demonstrating the four-colour theorem and NP-completeness. Eulerian and Hamiltonian path finders show the algorithmic difference between traversing every edge once versus every vertex once.

Permutation and subset enumeration visualisers animate the systematic generation of all arrangements and selections from a set, building intuition for n! and 2ⁿ growth rates. The Travelling Salesman Problem simulator compares nearest-neighbour heuristic, 2-opt, and branch-and-bound solutions on random point sets, showing the exponential hardness versus polynomial approximation trade-off. These models serve combinatorics courses, algorithm design students, and competitive programmers.

Each simulation in this category is built with accuracy and interactivity in mind. The underlying mathematical models are the same ones used in academic research and professional engineering — just made accessible through a web browser. Changing parameters in real time and observing the results is one of the most effective ways to build intuition for complex scientific and engineering concepts.