Ladder graph

Ladder graph
The ladder graph L8.
Vertices⁠⁠
Edges⁠⁠
Chromatic number⁠⁠
Chromatic index
PropertiesUnit distance
Hamiltonian
Planar
Bipartite
Notation⁠⁠
Table of graphs and parameters

In the mathematical field of graph theory, the ladder graph Ln is a planar, undirected graph with 2n vertices and 3n – 2 edges.[1]

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.[2][3]

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 .

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

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.

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

Circular ladder graph

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.[4] 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

Möbius ladder

Main article: Möbius ladder

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

Two views of the Möbius ladder M16 .

References

  1. ^ Weisstein, Eric W. "Ladder Graph". MathWorld.
  2. ^ Hosoya, H. and 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. ^ Chen, Yichao; Gross, Jonathan L.; Mansour, Toufik (September 2013). "Total Embedding Distributions of Circular Ladders". Journal of Graph Theory. 74 (1): 32–57. CiteSeerX 10.1.1.297.2183. doi:10.1002/jgt.21690. S2CID 6352288.

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