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https://hdl.handle.net/2142/20585

Description

Title

On Tait's color-tiling problem

Author(s)

Hu, Zhu-Xin

Issue Date

1996

Doctoral Committee Chair(s)

Weichsel, Paul M.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

In this thesis, we study a problem that generalizes a color tiling problem studied by the Scottish mathematician P. G. Tait in 1883, which has been of interest in China for many decades.

Let l, m, t be positive integers with $m\mid l$ and let $n\sb1,\ n\sb2,\...,\ n\sb{t}$ be nonnegative integers. We consider sequences (also called strings) with $n\sb1+n\sb2+\...+n\sb{t}+l$ positions in a line. Of these positions, $n\sb1$ are filled with tiles of color 1, $n\sb2$ are filled with tiles of color 2, $\...,$ $n\sb{t}$ are filled with tiles of color t, and the remaining l are left empty, indicated by 0. A segment of a string is a substring consisting of tiles or empty positions in contiguous positions. A solid segment is a segment containing no empty positions. An O-segment is a segment consisting of contiguous empty positions (like 00$\...$0). We say an O-segment is a maximal O-segment if it is not a part of a longer O-segment. A sequence is called a $(l,m;n\sb2,\...,n\sb{t})$-sequence if (a) the length of its longest solid segment is at least m, and (b) the length of each of its maximal O-segments is a multiple of m. An m-embedding permutation (m-EP) on a $(l,m;n\sb1,n\sb2,\... n\sb{t})$-sequence is a transformation that moves m contiguous pieces (without alternating their relative positions) into m contiguous empty positions such that the resulting sequence is also a $(l,m;n\sb1,n\sb2,\... n\sb{t})$-sequence. For any two $(l,m;n\sb1,n\sb2,\... n\sb{t})$-sequences X and Y, we define $d(X, Y)$ to be the minimum number of m-EPs to transform X into Y if it can be done so otherwise we define $d(X, Y)=\infty.$

In Chapter 1, we study the general t-color $(l,m;n\sb1,n\sb2,\...,n\sb{t})$-sequences. We obtained conditions with which the distance between any two $(l,m;n\sb1,n\sb2,\..., n\sb{t})$-sequences is bounded above by a linear function of $l+n\sb1+n\sb2+\...+n\sb{t}.$

In Chapter 2, we prove a long-standing conjecture. Let $X\sb0=(12)\sp{n}0\sp3,$ and $B(3, 3; n, n)=\{1\sp{n}2\sp{n}0\sp3,\ 2\sp{n}1\sp{n}0\sp3,\ 0\sp32\sp{n}1\sp{n}\}.$ C. Y. Chiang, in a paper in 1936, made the following conjecture: If n is an even integer $\ge$6, then for any $X\in B(3, 3; n, n),$ we have $d(X\sb0,X)=n+1.$

A consequence of the results of Chapter 2 is that Chiang's conjecture is true.

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