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path_._graph_theory [2014/02/09 03:17] nikolaj |
path_._graph_theory [2014/03/21 11:11] (current) |
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| ===== 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]] | + | [[Sequence]], [[Vertex degree]] |
