Let G be a finite connected simple graph and μ (G) be the Mycielskian of G. We show that for
connected graphs G and H, μ (G) is isomorphic to μ (H) if and only if G is isomorphic to H …

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least
one neighbor in each of the other color classes. A graph G which has a fall k-coloring is …

Given a proper coloring f of a graph G, a b-vertex in f is a vertex that is adjacent to every
color class but its own. It is a fall-coloring if every vertex is a b-vertex. The fall-spectrum of G …

Given a graph G, a (proper) k-coloring of G is a function f: V (G)→{1,..., k} such that f (u) ̸= f
(v), for every edge uv∈ E (G). Given a k-coloring f of a graph G, a vertex u∈ V (G) is a b …

For a simple connected undirected graph G, χ (G),\chi_f (G),\Psi_f (G), ϕ (G), ∂ Γ (G) and Ψ
(G) denote respectively the chromatic number, fall chromatic number (assuming that it exists …

The b-chromatic number of G, denoted by φ (G), is the maximum k for which G has a b-
coloring by k colors. A b-coloring of G by k colors is a proper k-coloring of the vertices of G …