Ladder graph

---

title: "Ladder graph" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["parametric-families-of-graphs", "planar-graphs"] description: "Planar, undirected graph with 2n vertices and 3n-2 edges" topic_path: "general/parametric-families-of-graphs" source: "https://en.wikipedia.org/wiki/Ladder_graph" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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

::data[format=table title="infobox graph"]

Field	Value
name	Ladder graph
image	[[Image: Ladder graph L8.svg
image_caption	The ladder graph L.
vertices
edges
chromatic_number
chromatic_index	\begin{cases}
3 & \text{if } n > 2 \ 2 & \text{if } n	2 \ 1 & \text {if } n = 1 \end{cases}
notation
properties	Unit distance
Hamiltonian
Planar
Bipartite
::

| name = Ladder graph | image = [[Image: Ladder graph L8.svg|120px]] | image_caption = The ladder graph L. | vertices = | edges = | automorphisms = | chromatic_number = | chromatic_index = \begin{cases} 3 & \text{if } n 2 \ 2 & \text{if } n = 2 \ 1 & \text {if } n = 1 \end{cases} |notation = | properties = Unit distance Hamiltonian Planar Bipartite

In the mathematical field of graph theory, the ladder graph L 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: .

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 n1) and chromatic index 3 (if n2).

The chromatic number of the ladder graph is 2 and its chromatic polynomial is (x-1)x(x^2-3x+3)^{(n-1)}.

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/ae/Ladder_graphs.svg" caption="The ladder graphs ''L''₁, ''L''₂, ''L''₃, ''L''₄ and ''L''₅."] ::

Image:Ladder graph L8 2COL.svg|The chromatic number of the ladder graph is 2.

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. ::figure[src="https://upload.wikimedia.org/wikipedia/commons/9/91/Ladder_rung_graphs.svg" caption="The ladder rung graphs ''LR''₁, ''LR''₂, ''LR''₃, ''LR''₄, and ''LR''₅."] ::

Circular ladder graph

Main article: Prism graph

The circular ladder graph CL**n 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: ::data[format=table] | [[File:Triangular prismatic graph.svg|100px]]CL3 | [[File:Cubical graph.svg|100px]]CL4 | [[File:Pentagonal prismatic graph.svg|100px]]CL5 | [[File:Hexagonal prismatic graph.svg|100px]]CL6 | [[File:Heptagonal prismatic graph.svg|100px]]CL7 | [[File:Octagonal prismatic graph.svg|100px]]CL8 | |---|---|---|---|---|---| ::

Möbius ladder

Main article: Möbius ladder

Connecting the four 2-degree vertices of a standard ladder graph crosswise creates a cubic graph called a Möbius ladder. ::figure[src="https://upload.wikimedia.org/wikipedia/commons/e/e3/Moebius-ladder-16.svg" caption="Two views of the Möbius ladder ''M''₁₆ ."] ::

References

References

1. "Ladder Graph".
2. [[Haruo Hosoya. Hosoya, H.]] and [[Frank Harary. Harary, F.]] "On the Matching Properties of Three Fence Graphs." J. Math. Chem. 12, 211-218, 1993.
3. Noy, M. and Ribó, A. "Recursively Constructible Families of Graphs." Adv. Appl. Math. 32, 350-363, 2004.
4. (September 2013). "Total Embedding Distributions of Circular Ladders". [[Journal of Graph Theory]].

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::