What is interesting about this is the serendipitous manner in which results build on each other: our result consisted in a partial solution to an original problem in combinatorics posed by the itinerant mathematician Paul Ërdos (posed in the mid 1960s), which we then used to partially solve a problem related to string indeterminates (also in this case working on previous results of Joel Helling [post]), which are related to genetics. Now, our work is being used to solve the problem of satellite allocation.
Our paper is on indeterminate strings, which are important for their applicability in bioinformatics. (They have been considered, for example, in Christodoulakis 2015 and Helling 2017.)
An interesting feature of indeterminate strings is the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimal alphabet size of indeterminates representing any given undirected graph corresponds to the size of the minimal clique cover of this graph. This paper first considers a related problem proposed in Helling 2017: characterize $latex\Theta_n(m)$, which is the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph on vertices and edges. We provide improvements to the known upper bound for . Helling 2017 also presents an algorithm which finds clique covers in polynomial time. We build on this result with an original heuristic for vertex sorting which significantly improves their algorithm’s results, particularly in dense graphs.
This work was the result of building on Helling 2017 (see this post) and of a year of research undertaken by Ryan McIntyre under my (Michael Soltys) supervision at the California State University Channel Islands.