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Ladder graph

Planar, undirected graph with 2n vertices and 3n-2 edges

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated August 11, 2026
Image credit is listed at the end of this article.

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 = Pn □ P2.

01Properties

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(x^{2}-3x+3)^{(n-1)}.

The ladder graphs L1, L2, L3, L4 and L5.
The ladder graphs L1, L2, L3, L4 and L5.

02Ladder rung graph

Sometimes the term "ladder graph" is used for the nP2 ladder rung graph, which is the graph union of n copies of the path graph P2.

The ladder rung graphs LR1, LR2, LR3, LR4, and LR5.
The ladder rung graphs LR1, LR2, LR3, LR4, and LR5.

03Circular 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:


CL3
CL4
CL5
CL6
CL7
CL8
Two views of the Möbius ladder M16 .
Two views of the Möbius ladder M16 .

04Möbius ladder

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

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Sources and credits

This article is adapted from the Wikipedia article “Ladder graph”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

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