L(2,1)-coloring and its related problems for Mycielskians of certain classes of graphs

In an [Formula: see text]-coloring [Formula: see text] of [Formula: see text] with span [Formula: see text], [Formula: see text] is a hole if there is no vertex [Formula: see text] in [Formula: see text] such that [Formula: see text]. An [Formula: see text]-coloring with span [Formula: see text] is said to be a no-hole coloring if it uses all the colors from [Formula: see text]. An [Formula: see text]-coloring of a graph [Formula: see text] is irreducible if color of no vertex can be decreased and yield another [Formula: see text]-coloring of [Formula: see text]. A graph [Formula: see text] is inh-colorable if there exists an irreducible no-hole coloring of [Formula: see text]. The inh-span of a graph [Formula: see text], denoted by [Formula: see text] is the smallest number [Formula: see text] such that there is an irreducible no-hole [Formula: see text]-coloring of [Formula: see text] with span [Formula: see text]. The maximum number of holes in [Formula: see text] is the maximum number of holes over all irreducible span colorings of [Formula: see text]. In this paper, we give an algorithm to define an [Formula: see text]-coloring for the Mycielskian [Formula: see text] of an arbitrary graph [Formula: see text]. We prove that the algorithm gives span colorings for Mycielskians of path [Formula: see text] ([Formula: see text]), cycle [Formula: see text] ([Formula: see text]), some classes of trees and concatenation of an arbitrary graph with trees. Also, we show that the Mycielskians for the above classes of graphs are inh-colorable and their inh-span is same as the [Formula: see text]-number. Further, we determine the maximum number of holes over all irreducible span colorings for Mycielskians of path [Formula: see text] ([Formula: see text]), cycle [Formula: see text] ([Formula: see text]), some classes of trees and concatenation of an arbitrary graph with trees.