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, Gmβ and Hmβ 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 Gmβ is maximal triangle-free, and when m is even, the addition of
exactly 2mβ edges makes the graph Hmβ maximal triangle-free. We
show that Gmβ is 4-chromatic and Hmβ is 3-chromatic for all m. Further,
we note some other properties of these graphs and compare with Mycielski's
construction.Comment: This is a first draft report about ongoing work on the Gr\"otzsch
Graph