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: .
By construction, the ladder graph L<sub>n</sub> is isomorphic to the grid graph G<sub>2,n</sub> 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 .
Sometimes the term "ladder graph" is used for the nP<sub>2</sub> ladder rung graph, which is the graph union of n copies of the path graph P<sub>2</sub>.
The circular ladder graph CL<sub>n</sub> 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. 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:
Connecting the four 2-degree vertices of a standard ladder graph crosswise creates a cubic graph called a Möbius ladder.