## Date of Submission

2-28-1970

## Date of Award

2-28-1971

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Doctoral Thesis

## Degree Name

Doctor of Philosophy

## Subject Name

Computer Science

## Department

Research and Training School (RTS)

## Supervisor

Rao, C. Radhakrishna (RTS-Kolkata; ISI)

## Abstract (Summary of the Work)

Graph thoory has becone such a well Imown and widely applicd subject with nuncrous applications in operations rescarch, coding thoory, gone thoory, physical and so oial sciences (to montion only a few), that it is not neccasary to give a goneral introduction to 1t. Instead wo give below a Burnary of the reaults containod in thia thosis chapterwisc.This thesis contains five chapters which are, morc or loss, indopendent of cach. other. In Chaptcr 1, we study the existence of locally restricted graphs, that is graphs having a proscribod property with given dogrecs. In Section 1.1, wc obtain necossary and sufficient conditions for the existonce of a p-connocted graph with givon degrocs for p= 3 and state two conjcctures in the gencral case. The concopt of k-factOr ablo soqucnces in introducc d in Section 1.2. A k-factor of a graph G is a partial graph of G in which every vertex has degrec k. A sequence is called k-factorablo (connectcd k-factorable) if there exists a graph with a k-fnctor (a graph wi th a connectod k-factor) with the givon degroc ac quenco. Wo obtnin a noccssary and aufficiont condition for a k-factorablo ncquonoe (* 2 2) to be connectod k-iectorable which turne out to bo indopondont of k. Sincc a connocted 2-factor is a Honiltonian cyclo, thfs condition ia neceesary and auffi- ciont for a 2-factorable soquence to be the degroe acqu- encc of a Haniltoni on graph. Wo prove that cvery k-fac- torable scqucnce (k > 2) io (k-2)-factorable, 2-facto- rable and 1-factornble. We furthor ahow that a 4-facto- rablo scqucncc is 3-fnctorable providod n (Ä‘, } ia a k-factorable scqucnee then there exista a is cven. If with dogrec scquence {a, } having a connected par tinl graph in which two vertocn have dogrece k-l and the reat havo Ä‘ogroon k. This provee, in particular, that ovory 2-factorablo aoquence ia realisable by a graph with a Haniltonian chain, Although, the goneral problon of characterising the k-factorable ec quencca in lcrt unanawored we precent two conjccturcs, onc of which for k = 2 wna nentioned by Prof. B. GrÃ¼nbaun at the Conbinatorics Conferenco, Colgnry 1969 and provo that the trutli of conjocture 1 inplics the truth of conjecturo 2 by proving that if {d1} and {d1-k} are graphical so is {d1-r} if and ue graphionl 30 PDF compreion, boR, web optntoton uning a vtamaad ovtaon copy of CISION POFO 3 o r < k. In Section 1.3, wo aolvo the following problon posod by A. M. Hobba in the book entitled, Recent Progress in Conbinatories! edited by W. T. Tutte [ 24]. For what values of ie thore a planar grapii on n vorticcs without loops or mltiple edges which has 12 vertices of de gree 5 and n- 12 verticos of de grce 6? Some othor related problens are alao solvod.Chapter : is strongly conneeted with the work of Renachandra Rao [ 19 ] who deternincd, anong othor things, the rangon of the numbor of cut vertices and cut edgen in an undirected gaph on n vortices withn edges.

## Control Number

ISILib-TH3

## 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/10263/2146

## Recommended Citation

Rao, Siddani Bhaskara Dr., "Contributions to the Theory of Directed and Undirected Graphs." (1971). *Doctoral Theses*. 301.

https://digitalcommons.isical.ac.in/doctoral-theses/301

## 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:28843347