Properly colored cycles in edge-colored graphs

In this thesis theoretical aspects were studied regarding edge-colored graphs.

‘Especially properly colored (PC) cycles have attracted much research attention in past decades,’ Ruonan Li says. ‘They are applied in various fields, such as social science and molecular biology. Our interest was from a theoretical point of view only.’ 

Ruonan succeeded in generalising the study of directed cycles towards edge-colored graphs. ‘It was a natural extension,’ she says. ‘There is a conjecture on disjoint cycles in directed graphs. I was very lucky to prove progress could be made this way.’

In later stages of the PhD project, Ruonan tried to prove disjoint PC cycles in edge-colored graphs.

‘After a long period of little progress, I finally received the acknowledgement from scholars in the field, and realized that I could apply my obtained methods to their questions,’ she says. ‘I was very happy, and also thankful. Some of their positively stated critical questions, were of enormous help and inspiration.’

PC theta graphs

The last chapter of her PhD work, focused on the difference between edge-colored graphs and directed graphs.

‘We found that the class of PC theta graphs plays an important role in characterizing the difference between edge-colored complete graphs and multipartite tournaments,’ Ruonan states. ‘Moreover, our study on large PC cycles is expected to contribute to a conjecture on PC Hamilton cycles’.

‘The suggestion from my supervisor, Professor Hajo Broersma, Professor of the Programmable Nanosystems Group, was very important in my project,’ Ruonan says ‘He suggested me to check the literature on topics carefully. His intuitive feeling proved right: some questions I tried to study have been done by other scholars, in a different expression.’

In the end of her PhD thesis, Ruonan notes that several new open questions exist in this exciting area of research. ‘Quite a number of new conjectures can be posed,’ she says. ‘I hope my work will attract the attention of many researchers.’

Future career

Regarding her future career, Ruonan favours an academic career in her home country China.

‘The prospects are good for my area of research, I believe’ she says. ‘My PhD project was a joint collaboration with North-western Polytechnical University in Xi’an, China, where large groups of researchers work on related subjects. Every two years, hundreds of scholars are gathered together by the National Conference on Graph Theory and Combinatorics. I am sure, as a PhD researcher, I can contribute to progress on mathematical subjects. During the project, I gained self-confidence. I’ve learnt that working patiently, and paying fair attention to details involved, lead to good results.'