Generalized Gr\"otzsch Graphs

The aim of this paper is to present a generalization of Gr\"otzsch graph. Inspired by structure of the Gr\"otzsch's graph, we present constructions of two families of graphs, $G_m$ and $H_m$ for odd and even values of $m$ respectively and on $n = 2m +1$ vertices. We show that each member of this family is non-planar, triangle-free, and Hamiltonian. Further, when $m$ is odd the graph $G_m$ is maximal triangle-free, and when $m$ is even, the addition of exactly $\frac{m}{2}$ edges makes the graph $H_m$ maximal triangle-free. We show that $G_m$ is 4-chromatic and $H_m$ is 3-chromatic for all $m$. Further, we note some other properties of these graphs and compare with Mycielski's construction.