Auparajita Krishna, a mathematician from Mohanlal Sukhadia University (MLSU) here has solved a statistical mystery (conjecture) on Graceful Tree, a subject under the Graph Theory in Discrete Mathematics, which reportedly have had  been pending for past five decades .

Krishnas paper “A note on some thoughts on the graceful tree conjecture” has recently featured in the Journal of Discrete Mathematical Sciences and Cryptography published by Taylor & Francis (UK) which acknowledge  the scientists effort in offering solution to a long standing problem on the subject.

Since it takes a few years for the original unconventional idea to be accepted, after Indexing, abstracting, reviewing in by major agencies like Zentralblatt (Germany), Mathematical Reviews (American) etc the research is expected to take some time before it reach a wider audience after which Krishna may be internationally acclaimed for her intensive study.

‘Regarding the graceful tree conjecture, for some reason it has been assumed to be true even without any proof. A generalized result means that ALL trees are graceful’ Krishna explained.

So far the efforts have yielded more and more specialized results meaning some new class of tree is discovered and labeling discovered if it exists. The failing of the assuming the conjecture to be true is also not explained but a blind faith persists in the mathematicians minds also that the conjecture is true.

Krishna said ‘ This superstitious belief to assume the conjecture to be true even without proof stems from the perception of equating the definition of tree with some other definitions of other traditional older branches of mathematics where definition alone predicts how every member of the defined object would behave or look like. But since graphs are pictorial representations, definition alone cannot predict how ALL trees would look like’.

Krishna  had attempted to directly solve this conjecture in 2004 also in paper titled ” A study of the major graph labeling of trees” which was published in Informatica (European journal)  and  this was one of the only two direct attempts in solving the graceful tree conjecture. Her achievement then too had been cited in a paper by mathematicians from Netherlands in a paper “Symmetries of Graceful trees” by Kraayenbrink, de Nijs, Vavic.

Finally zeroing in on the solution which is not confined by the often used binary logic of yes/no, true/false, for /against which characterises the human mind also. This is an unconventional answer, she said.

Significance of the research- A conjecture is an unsolved question which in the case of this paper is the about 50 year old Graceful Tree Conjecture. A tree is a type of graph , a picture with points and lines joined together.

This subject has prospered in the last couple of hundred years or so  only tremendously with the uniqueness being that Graph Theory is a core pure mathematics subject but is has wide applications in many subjects like Physics, Computer Sc, Psychology, Engineering etc. A particular area of Graph Theory is graph labeling where points and nodes are assigned labels/numbers which form a certain pattern. The labeled graphs are widely used in applications like communication networks, crystallography, radar etc. Graceful is a kind of labeling.

By: Geetha Sunil Pillai