# On Locally-Balanced 2-Partitions of Complete Multipartite Graphs

## DOI:

https://doi.org/10.51408/1963-0001## Keywords:

2-partition, Locally-balanced 2-partition, Equitable coloring, Complete multipartite graph## Abstract

A 2-partition of a graph G is a function f : V (G) →{White; Black}. A 2-partition f of a graph G is locally-balanced with an open neighborhood if for every ϑ ϵ V (G), ||{u ϵ NG(v): f (u) = White} |-| {u ϵ NG(v): f(u) = Black} ≤1; where NG(v) ={u ϵ V (G): uv ϵ E(G) }. A 2-partition f՛ 0 of a graph G is locallybalanced with a closed neighborhood if for every v ϵ V(G), ||{u ϵ NG(v): f՛ (u) = White} |-| {u ϵ NG[v]:; where NG[v] = NG(v)∪{v.} In this paper we give necessary and su±cient conditions for the existence of locally-balanced 2-partitions of complete multipartite graphs

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*Mathematical Problems of Computer Science*,

*49*, 7–17. https://doi.org/10.51408/1963-0001

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