In the mathematical field of graph theory, the ladder graph Ln is a planar, undirected graph with 2n vertices and 3n – 2 edges. The ladder graph can be obtained as the Cartesian product of two path graphs, one of which has only one edge: Ln,1 = Pn × P2 .

Properties

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.

Ladder rung graph

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.

Circular ladder 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. In symbols, CLn = Cn × P2. 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:

Möbius ladder

Connecting the four 2-degree vertices crosswise creates a cubic graph called a Möbius ladder.