Advertisement

IWOCA 2012: Combinatorial Algorithms pp 73-75

# Further Results on the Mycielskian of Graphs

• T. Kavaskar
Conference paper
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  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  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 , , , , . Several graph parameters, especially in domination theory, on Mycielskian of graphs have been discussed in , .

## Keywords

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

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Coekayne, E.J., Hedetniemi, S.T., Miller, D.J.: Properties of hereditary hypergraphs and middle graphs. Canad. Math. Bull. 21(4), 461–468 (1978)
2. 2.
Fisher, D.C., McKenna, P.A., Boyer, E.D.: Hamiltonicity, diameter, domination, packing, and biclique partitions of Mycielski’s graphs. Discrete Applied Mathematics 84(1-3), 93–105 (1998)
3. 3.
Chang, G.J., Huang, L., Zhu, X.: Circular chromatic numbers of Mycielski’s graphs. Discrete Mathematics 205(1-3), 23–37 (1999)
4. 4.
Halldorsson, M.M.: Approximating the minimum maximal independence number. Information Processing Letters 46(4), 169–172 (1993)
5. 5.
Larsen, M., Propp, J., Ullman, D.: The fractional chromatic number of Mycielski’s graphs. J. Graph Theory 19, 411–416 (1995)
6. 6.
Chellali, M., Maffray, F.: Dominator Colorings in Some Classes of Graphs. Graphs and Combinatorics 28, 97–107 (2012)
7. 7.
Mycielski, J.: Sur le coloriage des graphs. Colloq. Math. 3, 161–162 (1955)
8. 8.
Lin, W., Wu, J., Lam, P.C.B., Gu, G.: Several parameters of generalized Mycielskians. Discrete Applied Mathematics 154, 1173–1182 (2006)

## Copyright information

© Springer-Verlag Berlin Heidelberg 2012

## Authors and Affiliations

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

## Personalised recommendations

### Citepaper 