Induced cycles in triangle graphs

TL;DR

This paper characterizes graphs based on their triangle graphs being cycles, extends to $C_n$-free cases, and explores properties like being a tree, chordal, or perfect, including a conjecture on triangle packing and covering.

Contribution

It provides new characterizations of graphs with triangle graphs as cycles, trees, chordal, or perfect, and verifies a conjecture on triangle packing and covering.

Findings

Characterization of graphs with triangle graph as a cycle

Forbidden subgraph characterization for triangle graphs being a tree, chordal, or perfect

Verification of a conjecture on triangle packing and covering

Abstract

The triangle graph of a graph $G$, denoted by ${\cal T}(G)$, is the graph whose vertices represent the triangles ($K_3$ subgraphs) of $G$, and two vertices of ${\cal T}(G)$ 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 ${\cal T}(G)$ 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.