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  2. Double counting (proof technique) - Wikipedia

    en.wikipedia.org/wiki/Double_counting_(proof...

    If one has added a collection of edges already, so that the graph formed by these edges is a rooted forest with trees, there are () choices for the next edge to add: its starting vertex can be any one of the vertices of the graph, and its ending vertex can be any one of the roots other than the root of the tree containing the starting vertex.

  3. Cayley's formula - Wikipedia

    en.wikipedia.org/wiki/Cayley's_formula

    The complete list of all trees on 2,3,4 labeled vertices: = tree with 2 vertices, = trees with 3 vertices and = trees with 4 vertices. In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley.

  4. Nash-Williams theorem - Wikipedia

    en.wikipedia.org/wiki/Nash-Williams_Theorem

    In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and more generally forests) a graph can have:. A graph G has t edge-disjoint spanning trees iff for every partition , …, where there are at least t(k − 1) crossing edges (Tutte 1961, Nash-Williams 1961).

  5. Tree (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Tree_(graph_theory)

    In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. [1] A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.

  6. Incidence coloring - Wikipedia

    en.wikipedia.org/wiki/Incidence_coloring

    The bound is attained by trees and complete graphs: If G is a complete graph with at least two vertices then () = + If G is a tree with at least two vertices then () = + The main results were proved by Brualdi and Massey (1993).

  7. Method of analytic tableaux - Wikipedia

    en.wikipedia.org/wiki/Method_of_analytic_tableaux

    A graphical representation of a partially built propositional tableau. In proof theory, the semantic tableau [1] (/ t æ ˈ b l oʊ, ˈ t æ b l oʊ /; plural: tableaux), also called an analytic tableau, [2] truth tree, [1] or simply tree, [2] is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. [1]

  8. Kruskal's tree theorem - Wikipedia

    en.wikipedia.org/wiki/Kruskal's_tree_theorem

    The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.

  9. Fáry's theorem - Wikipedia

    en.wikipedia.org/wiki/Fáry's_theorem

    His work stressed the existence of a particular partition of the edges of a maximal planar graph into three trees known as a Schnyder wood. Tutte's spring theorem states that every 3-connected planar graph can be drawn on a plane without crossings so that its edges are straight line segments and an outside face is a convex polygon (Tutte 1963).