My area of interest in mathematics is combinatorics and graph theory. It is a beautiful and
relatively young branch of mathematics which really only evolved seriously in its own right back in around the 1900’s, although many ideas in the area have
existed much earlier. I very much appreciate the fact that ideas from almost all other areas of mathematics, such as analysis, topology, algebra, number theory, and probability, are involved in the area. Also, I love the fact that combinatorics and graph theory has applications to other subjects such as
computer science and mathematical
physics. For example, the latter includes the wonderful area of percolation theory.
Within combinatorics and graph theory, I am interested in many subareas. These include the following.
- Graph colourings. Roughly speaking, this area considers those problems that involve coloured graphs (either edge-coloured, vertex-coloured or total-coloured). One of the most famous results is Ramsey's theorem (1930).
- Extremal graph theory. This area studies problems which either maximises or minimises a certain property of a graph. Possibly the most famous result is Turán’s theorem (1941). More recently, many results have been proved with the application of Szemerédi's regularity lemma (1975).
- Probabilistic combinatorics. This area brings together the areas of probability theory and combinatorics. The area includes the theory of random graphs, and the so-called probabilistic method, where one can prove the existence of a prescribed mathematical object without actually constructing the object explicitly. A famous result is the Lovász local lemma, due to Erdős and Lovász (1975).
- Extremal set systems and hypergraphs. This area includes famous results such as
Sperner's theorem (1928), the Erdős-Ko-Rado theorem (1938; rediscovered in 1961), and the Kruskal-Katona theorem (1963, 1968), as well as many results in the theory of discrete isoperimetric inequalities.
- Combinatorial geometry. This is a rather broad area which can be described as the study of objects such as distances, angles and polytopes in Euclidean space.
I also see myself as a very keen problem solver in recreational mathematics. I have had many adventures here during my mathematical life, from taking part
in the British Mathematical Olympiad in high school, to marking a problem in the
International Mathematical Olympiad in 2002 (held in my hometown of Glasgow in Scotland!), to working at present
as a mentor for the United Kingdom Mathematics
Trust to talented high school students in the United Kingdom, who are enthusiastic about mathematical olympiads. I am interested in various
mathematical competitions such as the International Mathematical Olympiad and the
William Lowell Putnam Mathematical Competition.
My Erdős number is 2.
Below are some mathematically related things about me.
Below are some documents related to my teaching, at Sun Yat-sen University, and at The University of Memphis. I shall be adding more of these documents as they become available.
Published or accepted papers
- Degree powers in graphs with a forbidden forest (with Yongxin Lan, Zhongmei Qin and Yongtang Shi), submitted.
- Gallai-Ramsey number for K4 (with Colton Magnant, Akira Saito, Ingo Schiermeyer and Yongtang Shi), submitted.
- Turán function and H-decomposition problem for gem graphs (with Teresa Sousa), Discussiones Mathematicae Graph Theory, 38 (2018), 717-741.
- Highly connected subgraphs of graphs with given independence number (with Shinya Fujita and Amites Sarkar), European Journal of Combinatorics, 70 (2018), 212-231 (pdf, 406kb).
- On various (strong) rainbow connection numbers of graphs (with Lin Chen, Xueliang Li and Jinfeng Liu), The Australasian Journal of Combinatorics, 70 (2018), 137-156 (pdf, 271kb).
- Total rainbow connection of digraphs (with Hui Lei, Colton Magnant and Yongtang Shi), Discrete Applied Mathematics, 236 (2018), 288-305 (pdf, 415kb).
- Rainbow vertex connection of digraphs (with Hui Lei, Shasha Li and Yongtang Shi), Journal of Combinatorial Optimization, 35 (2018), 86-107 (pdf, 384kb).
- Spectral conditions for some graph properties (with Lihua Feng, Pengli Zhang, Weijun Liu, Minmin Liu and Yuqin Hu), Linear Algebra and its Applications, 524 (2017), 182-198 (pdf, 254kb).
- Decompositions of graphs into fans and single edges (with Teresa Sousa), Journal of Graph Theory, 85 (2017), 400-411 (pdf, 274kb).
- Incidence graphs constructed from t-designs (with Xu Yang, Weijun Liu and Lihua Feng), Applicable Analysis and Discrete Mathematics, 10 (2016), 457-478 (pdf, 348kb).
- The spectral radius of edge chromatic critical graphs (with Jianxiang Cao, Shifeng Ding, Lihua Feng and Weijun Liu), Linear Algebra and its Applications, 492 (2016), 78-88 (pdf, 254kb).
- Monochromatic clique decompositions of graphs (with Oleg Pikhurko and Teresa Sousa), Journal of Graph Theory, 80 (2015), 287-298 (pdf, 264kb).
- Monochromatic structures in edge-coloured graphs and hypergraphs - a survey (with Shinya Fujita and Colton Magnant), International Journal of Graph Theory and its Applications, 1 (2015), 3-56 (pdf, 392kb).
- Rainbow k-connection in dense graphs (with Shinya Fujita and Colton Magnant), Journal of Combinatorial Mathematics and Combinatorial Computing, 93 (2015), 33-52 (pdf, 235kb).
- Total rainbow k-connection in graphs (with Ângela Mestre and Teresa Sousa), Discrete Applied Mathematics, 174 (2014), 92-101 (pdf, 327kb).
- Rainbow connection for some families of hypergraphs (with Rui Pedro Carpentier, Manuel Silva and Teresa Sousa), Discrete Mathematics, 327 (2014), 40-50 (pdf, 208kb).
- Monochromatic Kr-decompositions of graphs (with Teresa Sousa), Journal of Graph Theory, 76 (2014), 89-100 (pdf, 188kb).
- Rainbow vertex k-connection in graphs (with Ângela Mestre and Teresa Sousa), Discrete Applied Mathematics, 161 (2013), 2549-2555 (pdf, 180kb).
- The balanced decomposition number of TK4 and series-parallel graphs (with Shinya Fujita), Discussiones Mathematicae Graph Theory, 33 (2013), 347-359 (pdf, 188kb).
- Further results on the balanced decomposition number (with Shinya Fujita), in Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, 202 (2010), 119-128 (pdf, 360kb).
- The balanced decomposition number and vertex connectivity (with Shinya Fujita), SIAM Journal on Discrete Mathematics, 24 (2010), 1597-1616 (pdf, 322kb).
- Highly connected coloured subgraphs via the regularity lemma (with Yury Person), Discrete Mathematics, 309 (2009), 6277-6287 (pdf, 259kb).
- Highly connected monochromatic subgraphs of multicoloured graphs (with Robert Morris and Noah Prince), Journal of Graph Theory, 61 (2009), 22-44 (pdf, 286kb).
- Highly connected multicoloured subgraphs of multicoloured graphs (with Robert Morris and Noah Prince), Discrete Mathematics, 308 (2008), 5096-5121 (pdf, 328kb).
- A new lower bound for the convex decomposition number of point sets in the plane, manuscript (pdf, 222kb).
- Highly connected monochromatic subgraphs of multicoloured graphs: addendum (with Robert Morris and Noah Prince), manuscript (pdf, 158kb).
Unfortunately, I have decided not to put up my PhD dissertation here, due to copyright reasons. The title is "Recent extremal problems in combinatorics" (2006). Send me an at firstname.lastname@example.org if you are interested.
- Highly connected subgraphs of graphs with given independence number (extended Abstract) (with Shinya Fujita and Amites Sarkar; short version of paper 21), in: Discrete Mathematics Days - JMDA16, Electronic Notes in Discrete Mathematics, 54 (2016), 103-108 (pdf, 217kb).
- Monochromatic Kr-decompositions of graphs (extended abstract) (with Teresa Sousa; short version of paper 8), Electronic Notes in Discrete Mathematics, 43 (2013), 121-127 (pdf, 128kb).
- Rainbow k-connection in dense graphs (extended abstract) (with Shinya Fujita and Colton Magnant; short version of paper 11), in: Proceedings of EuroComb '11, Electronic Notes in Discrete Mathematics, 38 (2011), 361-366 (pdf, 148kb).
Conference and Seminar Talks
Here are some slides to the talks that I have given at conferences and seminars.
- Degree powers in graphs with a forbidden forest (with Yongxin Lan, Zhongmei Qin and Yongtang Shi). Talk given at The Japanese Conference on Combinatorics and its Applications (JCCA 2018)
in Sendai, Sendai, Japan, 20 to 24 May 2018; Topology and Combinatorics seminar, Ajou University, Suwon, South Korea, 21 August 2018; and AMS Sectional Meeting (Special Session on Structural Graph Theory), University of Hawaii at Manoa, Honolulu, USA, 22 to 24 March 2019. These slides are from the JCCA 2018 (pdf, 447kb).
- Rainbow cycles through specified vertices. Talk given at Workshop on Colored Notions of Connectivity in Graphs, Nankai University, Tianjin, China, 29 to 31 May 2017 (as an invited speaker). A similar talk was also given at International Conference on Cycles in Graphs, Vanderbilt University, Nashville, USA, 30 May to 2 June 2012. These slides are from the workshop at Nankai University (pdf, 541kb).
- Highly connected subgraphs in sparse graphs (with Shinya Fujita and Amites Sarkar). Talk given at KAIST Discrete Math Seminar, KAIST, Daejeon, South Korea, 15 June 2017; The 16th KIAS Combinatorics Workshop, KIAS, Seoul, South Korea, 2 to 3 June 2017 (as an invited speaker); The Third Sino-Japan Symposium on Graph Theory, Combinatorics and their Applications, Northwestern Polytechnical University, Xi’an, China, 3 to 6 November 2016; 2016 International Conference on Graph Theory, Combinatorics and Applications, Zhejiang Normal University, Jinhua, China, 28 to 31 October 2016 (as an invited speaker); and The Japanese Conference on Combinatorics and its Applications (JCCA 2016) in Kyoto, Kyoto University, Kyoto, Japan, 21 to 25 May 2016. These slides are from the JCCA 2016 (pdf, 369kb).
- Connected subgraphs in edge-coloured graphs (with Shinya Fujita and Colton Magnant). Talk given at Topology and Combinatorics seminar, Ajou University, Suwon, South Korea, 7 June 2017; Simon Fraser University Discrete Mathematics Seminar, Simon Fraser University, Burnaby, Canada, 10 March 2015; and Workshop on Graph Theory and Combinatorial
Optimization, Nankai University, Tianjin, China, 14 to 15 November 2015 (as an invited speaker). These slides are from the seminar at SFU (pdf, 416kb).
- Rainbow connection in hypergraphs (with Rui Pedro Carpentier, Manuel Silva and Teresa Sousa). Talk given at Bordeaux Graph Workshop 2012, LaBRI, Université Bordeaux 1, Talence, France, 21 to 24 November 2012 (pdf, 792kb).
- Rainbow k-connection in dense graphs (with Shinya Fujita and Colton Magnant). Talk given at EuroComb '11, Eötvös Lorand University, Budapest, Hungary, 29 August to 2 September 2011 (pdf, 772kb).
- The balanced decomposition number of a graph (with Shinya Fujita). Talk given at 8th French Combinatorial Conference, Université de Paris XI - Sud, Orsay, France, 28 June to 2 July 2010; and 41st Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Florida Atlantic University, Boca Raton, USA, 8 to 12 March 2010. These slides are from the 8FCC (pdf, 489kb).
- Highly connected coloured subgraphs and the regularity lemma (with Yury Person). Talk given at Additive Combinatorics in Lisbon, Universidade
de Lisboa, Lisbon, Portugal, 21 to 24 June 2010 (pdf, 778kb).
- Highly connected subgraphs in multicoloured graphs (with Robert Morris and Noah Prince). Talk given at Final COMBSTRU Workshop, Universitat Politècnica de
Catalunya, Barcelona, Spain, 25 to 27 September 2006; Horizon of Combinatorics, Lake Balaton, Hungary, 16 to 22 July 2006; and PIMS-MITACS Summer School in Probability, University of British Columbia, Vancouver, Canada, 6 to 30 June 2005. These slides (reproduced from an old format) are from the Final COMBSTRU Workshop (pdf, 383kb).
Guangzhou Discrete Mathematics Seminar
At Sun Yat-sen University, I am currently a co-organiser of Guangzhou Discrete Mathematics Seminar. My co-organisers of this seminar series are Ping Hu, Chao Yang and Zanbo Zhang. Click on the link of the seminar series for further details.
International Journal of Graph Theory and its Applications
I am an editor-in-chief for a new journal called International Journal of Graph Theory and its Applications (IJGTA). The inaugural issue was launched in June 2015. We are aiming for this journal to be of high quality. If you would like to submit a (high quality!) paper, please click on the link to the journal for more information. Please also help to spread the word about the journal by telling your colleagues about it, at your institutions, at conferences and workshops, etc.
Essays (written during my undergraduate days)
- Ergodic theory and Szemerédi's theorem: my essay when I was doing Part III of the Mathematical Tripos at University of Cambridge (pdf, 321kb).
- Discrete isoperimetric inequalities: my essay in my final year at University of London (pdf, 465kb).
Some notes that I gave to my UKMT mentees (for olympiad training)
Here are some useful notes if you are still in secondary school (i.e., high school) and you are enthusiastic about olympiad mathematics. These notes are produced as I go along working as a mentor in the UKMT mentoring scheme.