Editing Combinatorics and Graph Theory (section)

== <span style="color: #FFFFFF;">Remembering</span> ==
* '''Combinatorics''' — The branch of mathematics studying counting, arrangement, and structure in finite and discrete settings.
* '''Permutation''' — An ordered arrangement of objects; P(n,k) = n!/(n-k)! arrangements of k items from n.
* '''Combination''' — An unordered selection; C(n,k) = n!/[k!(n-k)!] = "n choose k"; denoted ⁿCₖ or C(n,k).
* '''Binomial coefficient''' — C(n,k); coefficients in the expansion of (a+b)^n (Pascal's triangle).
* '''Pigeonhole principle''' — If n+1 objects are placed in n boxes, at least one box contains ≥ 2 objects; powerful despite simplicity.
* '''Inclusion-exclusion principle''' — |A ∪ B| = |A| + |B| - |A ∩ B|; generalized to n sets.
* '''Generating function''' — A formal power series encoding a combinatorial sequence; used to solve recurrences.
* '''Graph''' — A set of vertices V and edges E ⊆ V×V; the central object of graph theory.
* '''Degree (graph)''' — The number of edges incident to a vertex; handshaking lemma: Σdeg(v) = 2|E|.
* '''Path''' — A sequence of distinct vertices connected by edges.
* '''Cycle''' — A path that starts and ends at the same vertex.
* '''Tree''' — A connected acyclic graph; n vertices and n-1 edges.
* '''Euler path/circuit''' — A path/circuit traversing each edge exactly once; exists iff 0 or 2 vertices have odd degree.
* '''Hamiltonian path/cycle''' — A path/cycle visiting each vertex exactly once; NP-complete to decide in general.
* '''Graph coloring''' — Assigning colors to vertices so adjacent vertices have different colors; chromatic number χ(G).
* '''Four Color Theorem''' — Every planar graph can be 4-colored; proved in 1976 by Appel & Haken using computer.
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