// Workers AI · dad joke modeWhat did the Robertson graph say? I've got a lot of edges.
| Robertson graph | |
|---|---|
 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 | |
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]
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 graph is neither planar nor 1-planar.[6]
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.[7]
Algebraic properties
[edit]The Robertson graph is not a vertex-transitive graph; 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.[8]
The characteristic polynomial of the Robertson graph is
Gallery
[edit]
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
[edit]- ↑ Weisstein, Eric W. "Class 2 Graph". MathWorld.
- ↑ Weisstein, Eric W. "Robertson Graph". MathWorld.
- ↑ Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.
- ↑ Robertson, N. "The Smallest Graph of Girth 5 and Valency 4." Bull. Amer. Math. Soc. 70, 824-825, 1964.
- ↑ Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
- ↑ Pupyrev, Sergey (2025), "OOPS: Optimized One-Planarity Solver via SAT", in Dujmović, Vida; Montecchiani, Fabrizio (eds.), Proc. 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025), Leibniz International Proceedings in Informatics (LIPIcs), vol. 357, pp. 14:1–14:19, doi:10.4230/LIPIcs.GD.2025.14, ISBN 978-3-95977-403-1.
- ↑ Turcotte, J., & Yvon, S. (2021). 4-cop-win graphs have at least 19 vertices. Discrete Applied Mathematics, 301, 74-98.
- ↑ Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15, 2008.