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| 研究生： |
楊千儒 Chien-Ju Yang |
|---|---|
| 論文名稱： |
0和1構成的飽和矩陣、123-強制矩陣與12..k-置換趨避矩陣 Saturation, 123-forcing, and 12..k-permutation-avoiding of 0-1 matrices |
| 指導教授： |
蔡昇甫
Shen-Fu Tsai |
| 口試委員: | |
| 學位類別： |
碩士 Master |
| 系所名稱： |
理學院 - 數學系 Department of Mathematics |
| 論文出版年： | 2024 |
| 畢業學年度： | 112 |
| 語文別： | 英文 |
| 論文頁數： | 60 |
| 中文關鍵詞： | 0-1 矩陣 、趨避矩陣 、飽和 、強制矩陣 、置換趨避矩陣 |
| 外文關鍵詞： | 0-1 matrix, excluded matrix, saturation, forcing matrix, permutation-avoiding matrix |
| 相關次數： | 點閱：11 下載：0 |
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0-1矩陣是僅由0和1為元素構成的矩陣。令A為一個0-1矩陣，若存在任一A的子矩陣，透過將任意數量的元素1轉換成0可以使其與一0-1矩陣P相等，則稱此0-1矩陣A包含0-1矩陣P，且該子矩陣稱為一個P的複製。若0-1矩陣A不包含P，且將任一個元素0轉換成1會使得矩陣A包含P ，那麼我們稱A是一個P的飽和矩陣。取自[1]和[2]的研究結果，我們將介紹有最大和最小數量的1的Ik的飽和矩陣和Jk的飽和矩陣，並將一些證明改的更精簡。
另外，我們還會討論的是強制矩陣和置換趨避矩陣，這兩種都是行數和列數相同
的矩陣。令A為一個n × n的0-1矩陣，若A所包含的每個大小為n × n的置換矩陣皆包
含單位矩陣I3，那麼我們稱A是一個123-強制矩陣。若A所包含的每個n × n的置換矩陣
皆不包含k × k的單位矩陣Ik ，那麼A被稱為一個12 . . . k-置換趨避矩陣。我們會仔細釐
清[3]中對於有最大數量的1的123-強制矩陣的條件。最後，為了解決[1]中的一項未解問
題，對於12 . . . k-置換趨避矩陣，我們猜想在任意n × n的12 . . . k-置換趨避矩陣中，0的最小數量為((n-k+2)¦2)。
A 0-1 matrix consists of entries that are either 0 or 1. A 0-1 matrix A contains a 0-1
matrix P if A has a submatrix that can be made equal to P by changing any number
of 1-entries to 0-entries. This submatrix is called a copy of P in A. A 0-1 matrix A is
saturating for a 0-1 matrix P if A does not contain P, yet turning an arbitrary 0-entry of A into a 1-entry creates a copy of P in A. In the saturation problems, we introduce the results from [1] and [2] concerning the maximum and minimum numbers of 1-entries in matrices saturating for Ik and Jk, and succinctly rephrase some of the proofs. Another line of research focuses on square matrices. An n × n 0-1 matrix A is a 123-forcing matrix if every n × n permutation matrix in A contains the identity matrix I3. Conversely, an n × n 0-1 matrix A is 12..k-permutation-avoiding if none of n × n permutation matrix in A contains the identity matrix Ik. We clarify the characterization of 123-forcing matrices from [3] with the maximum number of 1-entries, and add missing parts to the original proof to make it complete. Finally, we conjecture that the minimum number of 0-entries in any n × n 12 . . . k-permutation-avoiding matrices is ((n-k+2)¦2), aiming to solve an open problem in [1].
[1] Richard A. Brualdi and Lei Cao. Pattern-avoiding (0,1)-matrices and bases of per- mutation matrices. Discrete Appl. Math., 304:196–211, 2021.
[2] Radoslav Fulek and Bal ́azs Keszegh. Saturation problems about forbidden 0-1 sub- matrices. SIAM J. Discrete Math., 35(3):1964–1977, 2021.
[3] Richard A. Brualdi and Lei Cao. 123-forcing matrices. Australas. J. Combin., 86:169– 186, 2023.
[4] Shen-Fu Tsai. Saturation of multidimensional 0-1 matrices. Discrete Math. Lett., 11:91–95, 2023.
[5] Richard A. Brualdi and Lei Cao. Blockers of pattern avoiding permutation matrices. Australas. J. Combin., 83:274–303, 2022.
