Induced cycles in triangle graphs

The triangle graph of a graph GG, denoted by T(G)T(G), is the graph whose vertices represent the triangles (K3K3 subgraphs) of GG, and two vertices of T(G)T(G) are adjacent if and only if the corresponding triangles share an edge. In this paper, we characterize graphs whose triangle graph is a cycle and then extend the result to obtain a characterization of CnCn-free triangle graphs. As a consequence, we give a forbidden subgraph characterization of graph GG for which T(G)T(G) is a tree, a chordal graph, or a perfect graph. For the class of graphs whose triangle graph is perfect, we verify a conjecture of the third author concerning packing and covering of triangles.