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We study the combinatorial properties of beta expansions. In particular we study those bases which admit points with finitely many or countably many expansions. This leads to interesting questions, such as what is the smallest base admitting a points with finitely many or countably many expansions. We also consider the dimension theory of the set of expansions for a typical point within a "natural" parameter space.

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This thesis contains the results of two investigations. The rst concerns the 1- factorizability of regular graphs of high degree. Chetwynd and Hilton proved in 1989 that all regular graphs of order 2n and degree 2n where > 1 2 ( p 7 1) 0:82288 are 1-factorizable. We show that all regular graphs of order 2n and degree 2n where is greater than the second largest root of 4x6 28x5 71x4 + 54x3 + 88x2 62x + 3 ( 0:81112) are 1-factorizable. It is hoped that in the future our techniques will yield further improvements to this bound. In addition our study of barriers in graphs of high minimum degree may have independent applications. The second investigation concerns partial latin squares that satisfy Hall's Condition. The problem of completing a partial latin square can be viewed as a listcolouring problem in a natural way. Hall's Condition is a necessary condition for such a problem to have a solution. We show that for certain classes of partial latin square, Hall's Condition is both necessary and su cient, generalizing theorems of Hilton and Johnson, and Bobga and Johnson. It is well-known that the problem of deciding whether a partial latin square is completable is NP-complete. We show that the problem of deciding whether a partial latin square that is promised to satisfy Hall's Condition is completable is NP-hard.

This thesis discusses various connections between codes over rings, in par- ticular linear Z4-codes and their Gray map images as orthogonal arrays and t-designs. It also introduces the connections between VC-dimension of binary codes and the strengths of the codes as orthogonal arrays. The second chapter concerns codes over rings. It is known that if we have a matrix A over a eld F, whose rows form a linear code, such that any t columns of A are linearly independent then A is an orthogonal array of strength t. I shall begin with generalising this theorem to any nite commutative ring R with identity. The case R = Z4 is particularly important, because of the Gray map, an isometry from Zn 4 (with Lee weight) to Z2n 2 (with Hamming weight). I determine further connections that exist between the strength of a linear code C over Z4 as an orthogonal array, the strength of its Gray map image as an orthogonal array and the minimum Hamming and Lee weights of its dual C?. I also nd that the strength of a binary code as an orthogonal array is less than or equal to its strong VC-dimension. The equality holds for linear binary codes. Furthermore, the lower bound is also determined for the strength of the Gray map image of any linear Z4-code. 4 Moreover, I show that if a code over any alphabet is an orthogonal array with a certain constraint then the supports of the codewords of some Hamming weight form a t-design. Furthermore, I prove that if a linear Z2- code satis es the t-mixture condition, then such a code is an orthogonal array of strength t. I then investigate if such connection also exists for non- linear Gray map images of linear Z4-codes, and prove that it does for some values t.

This thesis is an investigation into the algebraic number-theoretical properties of certain polynomial invariants of graphs and matroids. The bulk of the work concerns chromatic polynomials of graphs, and was motivated by two conjectures proposed during a 2008 Newton Institute workshop on combinatorics and statistical mechanics. The first of these predicts that, given any algebraic integer, there is some natural number such that the sum of the two is the zero of a chromatic polynomial (chromatic root); the second that every positive integer multiple of a chromatic root is also a chromatic root. We compute general formulae for the chromatic polynomials of two large families of graphs, and use these to provide partial proofs of each of these conjectures. We also investigate certain correspondences between the abstract structure of graphs and the splitting fields of their chromatic polynomials. The final chapter concerns the much more general multivariate Tutte polynomials—or Potts model partition functions—of matroids. We give three separate proofs that the Galois group of every such polynomial is a direct product of symmetric groups, and conjecture that an analogous result holds for the classical bivariate Tutte polynomial.