Extremal Graphs for Intersecting Triangles

It is known that for a graph on n vertices bn2/4c + 1 edges is sufficient for the existence of many triangles. In this paper, we determine the minimum number of edges sufficient for the existence of k triangles intersecting in exactly one common vertex. 1 Notation With integers n ≥ p ≥ 1, we let Tn,p denote the Turán graph, i.e., the complete p-partite graph on n vertices where each partite set has either bn/pc or dn/pe vertices and the edge set consists of all pairs joining distinct parts. Kr represents the complete graph on r vertices. For a graph G and a vertex x ∈ V (G), the neighborhood of x in G is denoted by NG(x) = {y ∈ V (G) : (x, y) ∈ E(G)}, or when clear, simply N(x). The degree of x in G, denoted by degG(x), or deg(x), is the size of NG(x). We use δ(G) and ∆(G) to denote the minimum and maximum degrees, respectively, in G. For a subset X ⊂ V (G), let G[X] denote the subgraph of G induced by X. A matching in G is a set of edges from E(G), no two of which share a common vertex, and the matching number of G, denoted by ν(G), is the maximum number of edges in a matching in G.