Robertson graph
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In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson.[2][3]
Robertson graph | |
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![]() The Robertson graph is Hamiltonian. | |
Named after | Neil Robertson |
Vertices | 19 |
Edges | 38 |
Radius | 3 |
Diameter | 3 |
Girth | 5 |
Automorphisms | 24 (D12) |
Chromatic number | 3 |
Chromatic index | 5[1] |
Book thickness | 3 |
Queue number | 2 |
Properties | Cage Hamiltonian |
Table of graphs and parameters |
The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964.[4] As a cage graph, it is the smallest 4-regular graph with girth 5.
It has chromatic number 3, chromatic index 5, diameter 3, radius 3 and is both 4-vertex-connected and 4-edge-connected. It has book thickness 3 and queue number 2.[5]
The Robertson graph is also a Hamiltonian graph which possesses 5,376 distinct directed Hamiltonian cycles.
The Robertson graph is one of the smallest graphs with cop number 4.[6]
Algebraic properties
Summarize
Perspective
The Robertson graph is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 24, the group of symmetries of a regular dodecagon, including both rotations and reflections.[7]
The characteristic polynomial of the Robertson graph is
Gallery
- The Robertson graph as drawn in the original publication.
- The chromatic number of the Robertson graph is 3.
- The chromatic index of the Robertson graph is 5.
References
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