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 path_._graph_theory [2014/02/09 03:18]nikolaj path_._graph_theory [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2014/02/09 03:26 nikolaj 2014/02/09 03:18 nikolaj 2014/02/09 03:17 nikolaj 2014/02/09 03:17 nikolaj 2014/02/08 21:47 nikolaj 2014/02/08 19:28 nikolaj old revision restored (2014/02/08 19:25) Next revision Previous revision 2014/02/09 03:26 nikolaj 2014/02/09 03:18 nikolaj 2014/02/09 03:17 nikolaj 2014/02/09 03:17 nikolaj 2014/02/08 21:47 nikolaj 2014/02/08 19:28 nikolaj old revision restored (2014/02/08 19:25) Line 1: Line 1: ===== Path . graph theory ===== ===== Path . graph theory ===== ==== Set ==== ==== Set ==== - | @#88DDEE: $V,E$ ... set | + | @#55CCEE: context ​    | @#55CCEE: $V,E$ ... set | - | @#FFBB00: $\langle V,​E,​\psi\rangle \in \mathrm{it}(E,​V)$ | + | @#FFBB00: definiendum ​| @#FFBB00: $\langle V,​E,​\psi\rangle \in \mathrm{it}(E,​V)$ | - | @#55EE55: $\langle V,​E,​\psi\rangle$ ... simple graph | + | @#55EE55: postulate ​  | @#55EE55: $\langle V,​E,​\psi\rangle$ ... simple graph | - | @#DDDDDD: $u,v\in V$ | + | @#DDDDDD: range       | @#DDDDDD: $u,v\in V$ | - | @#DDDDDD: $a$ ... sequence in $V$  | + | @#DDDDDD: range       | @#DDDDDD: $a$ ... sequence in $V$  | - | @#DDDDDD: $i\in\mathbb N$ | + | @#DDDDDD: range       | @#DDDDDD: $i\in\mathbb N$ | - | @#55EE55: $d(v)\neq 0$ | + | @#55EE55: postulate ​  | @#55EE55: $d(v)\neq 0$ | - | @#55EE55: $\exists a.\ \forall u,v.\ (\exists i.\ \{a_{i},​a_{i+1}\}=\{u,​v\}) \leftrightarrow (\{u,​v\}\dots\mathrm{edge})$ | + | @#55EE55: postulate ​  | @#55EE55: $\exists a.\ \forall u,v.\ (\exists i.\ \{a_{i},​a_{i+1}\}=\{u,​v\}) \leftrightarrow (\{u,​v\}\dots\mathrm{edge})$ | ==== Discussion ==== ==== Discussion ==== A path is a graph which can fully be described by a sequence of vertices. A path is a graph which can fully be described by a sequence of vertices. + === Theorems === + The only paths which are non-bipartite are cycles of odd order. ==== Parents ==== ==== Parents ==== === Subset of === === Subset of === [[Simple graph]], [[Connected graph]] [[Simple graph]], [[Connected graph]] - === Requirements ​=== + === Context ​=== [[Sequence]],​ [[Vertex degree]] [[Sequence]],​ [[Vertex degree]] 