Local coloring for the mycielskian of a graph

Abstract

Let G = (V, E) be a graph. A local coloring of a graph G of order at least 2 is a function c : V(G) -> N having the property that for each set 5 C V(G) with 2 < |S| < 3, there exist vertices uyv € S such that |c(ti) - c(v)| > m$) where m, is the size of the induced subgraph (S). The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by Xℓ{c)- The local chromatic number of G is= min{xℓ(c)}> where the minimum is taken over all local colorings c of G. Let G be a graph with vertex set V(G) = {V₁,v₂, ...The Mycielskian M(G) of a graph G can be constructed from the graph G by adding n + 1 new vertices {u,v₁,V₂,...,V_n} and then for t> 1 < i < n, joining v[ to u and all the neighbors of In this article, we discuss about some results and bounds on the local chromatic number of the Mycielskian construction of a graph G. Also we present a triangle-free graph G of any order n > 9 with Xi) = In addition, we suggest some conjectures and mention a problem for further investigation. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.