Induced cycles in triangle graphs

Abstract

The triangle graph of a graph G, denoted by T (G), is the graph whose vertices represent the triangles (K 3 subgraphs) of G, and two vertices of 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 C n -free triangle graphs. As a consequence, we give a forbidden subgraph characterization of graphs G for which 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.

Triangle Packings and Transversals of Some $$K_{4}$$ K 4 -Free Graphs

“…As noticed in the proof of Lemma 8, an induced C 4 in a triangle graph T (G) corresponds to a W 4 in G. It is easy to see that a similar situation occurs for C 5 (see also [29]):…”

“…Note that if G is not K 4 -free, then an induced C 5 in T (G) may also correspond to a K 5 in G. For general C n , the situation becomes even more complicated. Nevertheless, Lakshmanan et al [29] provided a forbidden subgraph characterization of graphs with C n -free triangle graphs, for any specified n ≥ 3.…”