Robertson graph

{{infobox graph

| name = Robertson graph

| image = 240px

| image_caption = The Robertson graph is Hamiltonian.

| namesake = Neil Robertson

| vertices = 19

| edges = 38

| automorphisms = 24 (D₁₂)

| girth = 5

| diameter = 3

| radius = 3

| chromatic_number = 3

| chromatic_index = 5{{MathWorld|urlname=Class2Graph|title=Class 2 Graph}}

| properties = Cage
Hamiltonian

|book thickness=3|queue number=2}}

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.{{MathWorld|urlname=RobertsonGraph|title=Robertson Graph}}Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.

The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964.Robertson, N. "The Smallest Graph of Girth 5 and Valency 4." Bull. Amer. Math. Soc. 70, 824-825, 1964. 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.Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018

The Robertson graph is also a Hamiltonian graph which possesses {{formatnum:5376}} distinct directed Hamiltonian cycles.

The Robertson graph is one of the smallest graphs with cop number 4.Turcotte, J., & Yvon, S. (2021). 4-cop-win graphs have at least 19 vertices. Discrete Applied Mathematics, 301, 74-98.