Examples and Properties
Edge-transitive graphs include any complete bipartite graph, and any symmetric graph, such as the vertices and edges of the cube. Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. The Gray graph is an example of a graph which is edge-transitive but not vertex-transitive. All such graphs are bipartite, and hence can be colored with only two colors.
An edge-transitive graph that is also regular, but not vertex-transitive, is called semi-symmetric. The Gray graph again provides an example. Every edge-transitive graph must be bipartite and either semi-symmetric or biregular.
Read more about this topic: Edge-transitive Graph
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