| Ladder graph | |
| Chromatic Index: | \begin{cases} 3&ifn>2\ 2&ifn=2\ 1&ifn=1\end{cases} |
| Properties: | Unit distance Hamiltonian Planar Bipartite |
In the mathematical field of graph theory, the ladder graph is a planar, undirected graph with vertices and edges.
The ladder graph can be obtained as the Cartesian product of two path graphs, one of which has only one edge: .[1] [2]
By construction, the ladder graph Ln is isomorphic to the grid graph G2,n and looks like a ladder with n rungs. It is Hamiltonian with girth 4 (if n>1) and chromatic index 3 (if n>2).
The chromatic number of the ladder graph is 2 and its chromatic polynomial is
(x-1)x(x2-3x+3)(n-1)
Sometimes the term "ladder graph" is used for the n × P2 ladder rung graph, which is the graph union of n copies of the path graph P2.
See main article: Prism graph. The circular ladder graph CLn is constructible by connecting the four 2-degree vertices in a straight way, or by the Cartesian product of a cycle of length n ≥ 3 and an edge.[3] In symbols, . It has 2n nodes and 3n edges.Like the ladder graph, it is connected, planar and Hamiltonian, but it is bipartite if and only if n is even.
Circular ladder graph are the polyhedral graphs of prisms, so they are more commonly called prism graphs.
Circular ladder graphs:
See main article: Möbius ladder. Connecting the four 2-degree vertices crosswise creates a cubic graph called a Möbius ladder.