Equitable Colorings Of Corona Multiproducts Of Graphs

Hanna Furmánczyk 1 , Marek Kubale 2 ,  and Vahan V. Mkrtchyan 3
  • 1 Institute of Informatics, University of Gd√°nsk Wita Stwosza 57, 80-952 , Gd√°nsk, Poland
  • 2 Department of Algorithms and System Modelling Gd√°nsk University of Technology Narutowicza 11/12, 80-233 , Gd√°nsk, Poland
  • 3 Department of Informatics and Applied Mathematics Yerevan State University, , Yerevan, Armenia

Abstract

A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the numbers of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and denoted by ūĚúí=(G). It is known that the problem of computation of ūĚúí=(G) is NP-hard in general and remains so for corona graphs. In this paper we consider the same model of coloring in the case of corona multiproducts of graphs. In particular, we obtain some results regarding the equitable chromatic number for the l-corona product G ‚ó¶l H, where G is an equitably 3- or 4-colorable graph and H is an r-partite graph, a cycle or a complete graph. Our proofs are mostly constructive in that they lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of G. Moreover, we confirm the Equitable Coloring Conjecture for corona products of such graphs. This paper extends the results from [H. Furm√°nczyk, K. Kaliraj, M. Kubale and V.J. Vivin, Equitable coloring of corona products of graphs, Adv. Appl. Discrete Math. 11 (2013) 103‚Äď120].

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