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 ( x − 1 ) x ( x 2 − 3 x + 3 ) ( n − 1 ) {\displaystyle (x-1)x(x^{2}-3x+3)^{(n-1)}} .
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
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.
References
Weisstein, Eric W. "Ladder Graph". MathWorld. /wiki/Eric_W._Weisstein ↩
Hosoya, H. and Harary, F. "On the Matching Properties of Three Fence Graphs." J. Math. Chem. 12, 211-218, 1993. /wiki/Haruo_Hosoya ↩
Noy, M. and Ribó, A. "Recursively Constructible Families of Graphs." Adv. Appl. Math. 32, 350-363, 2004. ↩
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. /wiki/Toufik_Mansour ↩