Graphs which are edge-locally \(C_n\) (Q2702760)

The neighbourhood of an edge \(e\) of a graph \(G\) is the set of vertices which are adjacent to the end vertices of \(e\) and are distinct from them. If this set induces a subgraph isomorphic to a circuit \(C_n\) of length \(n\), the graph \(G\) is called edge-locally \(C_n\). Analogously a vertex-locally \(C_n\) graph is defined. The edge-locally \(C_n\) graphs containing circuits are characterized, using the property of being vertex-locally \(C_n\). The main result states that an edge-locally \(C_n\) graph exists if and only if \(n = 3\) or \(n\) is even, \(n > 4\), and that for each \(n \geq 8\) there are infinitely many connected edge-locally \(C_n\) graphs. A construction of such graphs is described. Interconnections with group theory are mentioned.