In graph theory, the Henson graphGi is an undirected infinite graph, the unique countable homogeneous graph that does not contain an i-vertex clique but that does contain all Ki-free finite graphs as induced subgraphs. For instance, G3 is a triangle-free graph that contains all finite triangle-free graphs.
These graphs are named after C. Ward Henson, who published a construction for them (for all i ≥ 3) in 1971. [1] The first of these graphs, G3, is also called the homogeneous triangle-free graph or the universal triangle-free graph.
To construct these graphs, Henson orders the vertices of the Rado graph into a sequence with the property that, for every finite set S of vertices, there are infinitely many vertices having S as their set of earlier neighbors. (Only the Rado graph has such a sequence.) He then defines Gi to be the induced subgraph of the Rado graph formed by removing the final vertex (in the sequence ordering) of every i-clique of the Rado graph. [1]
With this construction, each graph Gi is an induced subgraph of Gi + 1, and the union of this chain of induced subgraphs is the Rado graph itself. Because each graph Gi omits at least one vertex from each i-clique of the Rado graph, there can be no i-clique in Gi.
Any finite or countable i-clique-free graph H can be found as an induced subgraph of Gi by building it one vertex at a time, at each step adding a vertex whose earlier neighbors in Gi match the set of earlier neighbors of the corresponding vertex in H. That is, Gi is a universal graph for the family of i-clique-free graphs.
Because there exist i-clique-free graphs of arbitrarily large chromatic number, the Henson graphs have infinite chromatic number. More strongly, if a Henson graph Gi is partitioned into any finite number of induced subgraphs, then at least one of these subgraphs includes all i-clique-free finite graphs as induced subgraphs. [1]
Like the Rado graph, G3 contains a bidirectional Hamiltonian path such that any symmetry of the path is a symmetry of the whole graph. However, this is not true for Gi when i > 3: for these graphs, every automorphism of the graph has more than one orbit. [1]