Concept
In graph theory, a locally linear graph is an undirected graph in which every edge belongs to exactly one triangle. Equivalently, for each vertex of the graph, its neighbors are each adjacent to exactly one other neighbor, so the neighbors can be paired up into an induced matching. Locally linear graphs have also been called locally matched graphs. Their triangles form the hyperedges of triangle-free 3-uniform linear hypergraphs and the blocks of certain partial Steiner triple systems, and the locally linear graphs are exactly the Gaifman graphs of these hypergraphs or partial Steiner systems. Many constructions for locally linear graphs are known. Examples of locally linear graphs include the triangular cactus graphs, the line graphs of 3-regular triangle-free graphs, and the Cartesian products of smaller locally linear graphs. Certain Kneser graphs, and certain strongly regular graphs, are also locally linear. The question of how many edges locally linear graphs can have is one of the formulations of the Ruzsa–Szemerédi problem. Although dense graphs can have a number of edges proportional to the square of the number of vertices, locally linear graphs have a smaller number of edges, falling short of the square by at least a small non-constant factor. The densest planar graphs that can be locally linear are also known. The least dense locally linear graphs are the triangular cactus graphs. The friendship graphs, graphs formed by gluing together a collection of triangles at a single shared vertex, are locally linear. They are the only finite graphs having the stronger property that every pair of vertices (adjacent or not) share exactly one common neighbor. More generally every triangular cactus graph, a graph formed by gluing triangles at shared vertices without forming any additional cycles, is locally linear. Locally linear graphs may be formed from smaller locally linear graphs by the following operation, a form of the clique-sum operation on graphs.
