A Discourse on Three Combinatorial Diseases

Frank Harary in his article [2] was the first to speak of combinatorial diseases (actually, he spoke of graphical diseases). His original diseases were the four-colour disease, the hamiltonian disease and the reconstruction disease. Shortly thereafter, Read and Corneil [9] have discussed the graph isomorphism disease, and somewhat later, Huang, Kotzig and Rosa [4] spoke of a labelling disease. When does a combinatorial problem become a disease? Certainly the extreme ease of formulating the problem has something to do with it: most identified ”diseases” are understandable to undergraduates or even to good high school students. They are highly contagious, and so they attract the attention of not only professional mathematicians but also of scores of layman mathematicians. In what follows I would like to discuss briefly three of my favourite diseases: the labelling disease, the Erdös-Faber-Lovász disease, and Buratti’s problem.