Characterization and chromatic number of triangle-free graphs with diameter 2
For graph theorists, this provides a corrected characterization and new bounds on chromatic numbers for a specific class of graphs, but the contribution is incremental.
The paper corrects a flaw in a known characterization of 2-self-centered graphs (triangle-free graphs with diameter 2 that are not stars) and uses the corrected characterization to prove results about the chromatic number of such graphs.
In this paper, we consider triangle-free graphs with diameter 2. If a triangle-free graph $G$ with diameter 2 is not isomorphic to a star, then the radius of $G$ is also 2, where such a graph is also called a $2$-self-centered graph. Shekarriz et al. [A characterization for 2-self-centered graphs, Discuss. Math. Graph Theory 38 (2018), 27--37.] gave a characterization of 2-self-centered graphs. However, there is a slight flaw in their characterization. Thus, in this paper, we modify it and prove an accurate characterization of those graphs. Furthermore, by using our characterization, we prove some results concerning the chromatic number of triangle-free graphs with diameter 2.