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# Category Archives: open problems

## A small fact about the Petersen graph

The Petersen graph has 10 vertices and 15 edges, and the complete graph on 10 vertices has 45 edges. However, Allen Schwenk and (independently) O. P. Lossers (Jack van Lint’s problem-solving seminar in Eindhoven) showed that you can’t partition the … Continue reading

## Algebraic properties of chromatic polynomials, 2

My paper with Kerri Morgan on algebraic properties of chromatic roots (described here) has just appeared in the Electronic Journal of Combinatorics: you can find it here. I won’t say any more about it, except to pose a challenge which … Continue reading

Posted in open problems, Uncategorized
Tagged algebraic integer, chromatic polynomial, Galois group
2 Comments

## Automorphism groups of transformation semigroups

I have been in the transformation semigroups game for nearly ten years now, but I still feel that I am finding my feet. Here is apparently a huge difference between permutation groups and transformation semigroups, one which is still not … Continue reading

Posted in doing mathematics, open problems, Uncategorized
Tagged automorphism, primitive groups, synchronization
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## Problems

I have begun the long job of updating my collection of open problems. I would appreciate any help! My St Andrews problems are here, and are in pretty good shape (there are only 14 of them so this wasn’t a … Continue reading

## Discrete mathematics in Derby

This week I have been at a conference on “Theoretical and Computational Discrete Mathematics” at the University of Derby, under the auspices of the Institute for Mathematics and its Applications. The University of Derby was founded as the Derby Diocesan … Continue reading

## The power graph yet again

Five years ago, I posted a short update on the power graph of a group. Now, finally, the paper resulting from this has appeared on the arXiv; my coauthors are Ghodratollah Aalipour, Saieed Akbari, Reza Nikandish and Farzad Shaveisi. I … Continue reading

## Circular repeated-measurements designs

My first paper in a real statistical journal has just been almost accepted (just a bit of re-formatting …) The paper is entitled “On optimality and construction of circular repeated-measurements designs”, the other authors are R. A. Bailey, K. Filipiak, J. Kunert and A. Markiewicz. … Continue reading