Today was supposed to be a holiday as it is Kanakadaasa Jayanthi. While the rest of my classmates (except my project mate Anjali) had two precious days off, tomorrow being a Sunday, I had to go to college to work on my project on graph theory. At first, I felt a little lethargic as I didn't know what we would be discussing about with my guide; I had barely enough time to complete my assignments and the like and reading the paper that I was assigned needed a lot of time at a stretch. It has cases and subcases and subsubcases and few of them aren't even explicitly mentioned.

We started off by clearing Anjali's doubts, which involved line graphs. I was completely disinterested in line graphs. Most of the graph theory that I've learnt in my second semester isn't something that I recall with fondness. While sir had asked me a while back to work on two problems, I hadn't gotten around to even thinking about it as I felt I had to finish reading the paper first to get an idea of what techniques I might have to apply. So after Anjali's doubt was cleared and she was asked to work on a problem, I was asked as to why the proof of a similar theorem fails in my case. I started to think about it. Sir said he would take a walk and get back. Anjali and I were in his cabin, and we decided to go to the first year MSc math class, which had just been cleaned. We then silently began working on our problems independently.

I started to apply the technique of a known theorem about a certain class of graphs to my problem: P3 union K2 free graphs, a super class of the class of graphs already dealt with. I found where the exact same technique would fail and was glad I could now answer sir's question. He still hadn't returned from his walk. I then began to see if I could still obtain an upper bound for the chromatic number of the graphs I was now dealing with in terms of maximum clique size omega. I found that with few minor changes, I could. But this left me with a doubt as I thought he said this problem was not solved. Then how could it have been solved so easily by me?

Sir then came back, and asked me to prove the theorem that we had previously discussed on the board. I did that as neatly and concisely as I could. I drew a vertical line to separate it from my next theorem (this I have picked up from Choudum sir's way of handling the board) and proceeded to prove how I can still apply the same technique to the new problem, with a few minor changes. To my surprise I was able to clear any questions that sir had, and then it seemed that what I had done actually could be right. Sir treated it like as it if was correct and proceeded to ask me the next question: chromatic bound for P5 free graphs in terms of omega. But I said, "So what about the one we proved just now, has it been solved before?". To which he replied, "I do not know if it has been solved, but if it has they might have just let it go as well, without publishing it". I didn't quite know what to make of that, as I was happy I was able to prove something that sir did not know was proved or not; I was happy regardless the fact that it might already be proved as I had done some genuine research now.

Sir then said that the P5 free problem had definitely not been solved, for many years that too, so solving this would be a real contribution to graph theory. He wanted to proceed in the same manner as before, but I spotted that we couldn't do that, as P5 was connected. We still tried going about in a similar manner, trying to partition the vertices in such a way that it was advantageous to us. We reached a certain point and got stuck, and sir pointed out that that was why few people add diamond free graphs as well as that is easier to deal with. We kept thinking about how to solve the problem but we were unable to get past that block. Sir then said that we can think about this later, and said that he had a feeling that it might be possible to solve the problem this way.

All in all, I felt really happy and excited. Doing research is fun! It started off being a little boring today, but it goes to show that if you ask the right questions and persevere, you never know when things can take a turn!

We started off by clearing Anjali's doubts, which involved line graphs. I was completely disinterested in line graphs. Most of the graph theory that I've learnt in my second semester isn't something that I recall with fondness. While sir had asked me a while back to work on two problems, I hadn't gotten around to even thinking about it as I felt I had to finish reading the paper first to get an idea of what techniques I might have to apply. So after Anjali's doubt was cleared and she was asked to work on a problem, I was asked as to why the proof of a similar theorem fails in my case. I started to think about it. Sir said he would take a walk and get back. Anjali and I were in his cabin, and we decided to go to the first year MSc math class, which had just been cleaned. We then silently began working on our problems independently.

I started to apply the technique of a known theorem about a certain class of graphs to my problem: P3 union K2 free graphs, a super class of the class of graphs already dealt with. I found where the exact same technique would fail and was glad I could now answer sir's question. He still hadn't returned from his walk. I then began to see if I could still obtain an upper bound for the chromatic number of the graphs I was now dealing with in terms of maximum clique size omega. I found that with few minor changes, I could. But this left me with a doubt as I thought he said this problem was not solved. Then how could it have been solved so easily by me?

Sir then came back, and asked me to prove the theorem that we had previously discussed on the board. I did that as neatly and concisely as I could. I drew a vertical line to separate it from my next theorem (this I have picked up from Choudum sir's way of handling the board) and proceeded to prove how I can still apply the same technique to the new problem, with a few minor changes. To my surprise I was able to clear any questions that sir had, and then it seemed that what I had done actually could be right. Sir treated it like as it if was correct and proceeded to ask me the next question: chromatic bound for P5 free graphs in terms of omega. But I said, "So what about the one we proved just now, has it been solved before?". To which he replied, "I do not know if it has been solved, but if it has they might have just let it go as well, without publishing it". I didn't quite know what to make of that, as I was happy I was able to prove something that sir did not know was proved or not; I was happy regardless the fact that it might already be proved as I had done some genuine research now.

Sir then said that the P5 free problem had definitely not been solved, for many years that too, so solving this would be a real contribution to graph theory. He wanted to proceed in the same manner as before, but I spotted that we couldn't do that, as P5 was connected. We still tried going about in a similar manner, trying to partition the vertices in such a way that it was advantageous to us. We reached a certain point and got stuck, and sir pointed out that that was why few people add diamond free graphs as well as that is easier to deal with. We kept thinking about how to solve the problem but we were unable to get past that block. Sir then said that we can think about this later, and said that he had a feeling that it might be possible to solve the problem this way.

All in all, I felt really happy and excited. Doing research is fun! It started off being a little boring today, but it goes to show that if you ask the right questions and persevere, you never know when things can take a turn!

## 1 comment:

Check this out

https://www.gerad.ca/~alainh/P5DMA.pdf

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