# Decomposition and Packing of Cycles of a Given Length in a Complete Graph.

## Date of Submission

December 1994

## Date of Award

Winter 12-12-1995

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

## Subject Name

Computer Science

## Department

Applied Statistics Unit (ASU-Kolkata)

## Supervisor

Roy, Bimal Kumar (ASU-Kolkata; ISI)

## Abstract (Summary of the Work)

Existence of an edge-disjoint collection of cycles of given length m, which partition a complete undirected graph Kn of order n, depends on both m &n. In 1847, T. P. Kirkman(4] determined the spectrum of 3-cycle systems(the set of all n such that a 3-cycle system of order n exists). In 1892, spectrum problem for n-cycle systems of Kn wias settled[7). Kotzig[5] in 1965 and Rosa[10] in 1966 determined the spectrum of m-cycle systems, for all even m. But the general problem of packing is still unsolved.In this work, we have handled odd-cycle systems and even-cycle systems separately. It has been known that the spectrum of 3-cycle systems is precisely the set of all n = i or 3 (inod 6). We have considered those n where n = 0 or 2 or 5 (mod 6). Our objective is to pack largest possible no, of 3-cycles in a complete graph where no. of nodes is of the form 6t, 6t +2 or 6t +5. The packing problem in case of 4-cycle systems is handled in a different way. The spectrum of 4-cycle systems is precisely the set of all n= !(mod 8). In this work, we have considered the prolblem of packing 4-cycles in a complete graph of even order. A construction, that produces an edge disjoint collection of largest possible no. of 4-cycles from a complete graph of even order, is given. It has also been shown that this construction is cquivalent to Spouse-avoiding variant of Oberwolfach Problem proposed by IHuang, Kotzig and Rosa[3] in 1979.

## Control Number

ISI-DISS-1994-16

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/6191

## Recommended Citation

Chakraborty, Sharmistha, "Decomposition and Packing of Cycles of a Given Length in a Complete Graph." (1995). *Master’s Dissertations*. 209.

https://digitalcommons.isical.ac.in/masters-dissertations/209

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843232