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Further Results on the Mycielskian of Graphs

  • T. Kavaskar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7643)

Introduction

All graphs considered here are finite, undirected, connected and non-trivial. In the mid 20 th century there was a question regarding, triangle-free graphs with arbitrarily large chromatic number. In answer to this question, Mycielski [7] developed an interesting graph transformation as follows: For a graph G = (V,E), the Mycielskian of G is the graph μ(G) with vertex set consisting of the disjoint union V ∪ V′ ∪ {u}, where V′ = {x′:x ∈ V} and edge set E ∪ {xy:xy ∈ E} ∪ {xu:x′ ∈ V′}. We call x′ the twin of x in μ(G) and vice versa and u, the root of μ(G). We can define the iterative Mycielskian of a graph G as follows: μ m (G) = μ(μ m − 1(G)), for m ≥ 1. Here μ 0(G) = G. It is well known [7] that if G is triangle free, then so is μ(G) and that the chromatic number χ(μ(G)) = χ(G) + 1. There had been several papers on Mycielskian of graphs. Few of the references are [2], [3], [5], [7], [8]. Several graph parameters, especially in domination theory, on Mycielskian of graphs have been discussed in [2], [8].

Keywords

Mycielskian of a graph acyclic chromatic number dominator chromatic number independent domination number 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • T. Kavaskar
    • 1
  1. 1.Department of MathematicsBharathidasan UniversityTiruchirappalliIndia

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