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2. Domination of the rectangular queen's graph

Author

Bozóki, Sándor, Gál, Péter, Marosi, István, and Weakley, William D.

Subjects
Mathematics - Combinatorics
Abstract

The queen's graph $Q_{m \times n}$ has the squares of the $m \times n$ chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal of the board. A set $D$ of squares of $Q_{m \times n}$ is a dominating set for $Q_{m \times n}$ if every square of $Q_{m \times n}$ is either in $D$ or adjacent to a square in $D$. The minimum size of a dominating set of $Q_{m \times n}$ is the domination number, denoted by $\gamma(Q_{m \times n})$. Values of $\gamma(Q_{m \times n}), \, 4 \leq m \leq n \leq 18, \,$ are given here, in each case with a file of minimum dominating sets (often all of them, up to symmetry) in an online appendix at https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i4p45/HTML. In these ranges for $m$ and $n$, monotonicity fails once: $\gamma(Q_{8 \times 11}) = 6 > 5 = \gamma(Q_{9 \times 11}) = \gamma(Q_{10 \times 11}) = \gamma(Q_{11 \times 11})$. Lower bounds on $\gamma(Q_{m \times n})$ are given. In particular, if $m \leq n$ then $\gamma(Q_{m \times n}) \geq \min \{ m, \lceil (m+n-2)/4 \rceil \}$. A set of squares is independent if no two of its squares are adjacent. The minimum size of an independent dominating set of $Q_{m \times n}$ is the independent domination number, denoted by $i(Q_{m \times n})$. Values of $i(Q_{m \times n}), \, 4 \leq m \leq n \leq 18, \,$ are given here, in each case with some minimum dominating sets. In these ranges for $m$ and $n$, monotonicity fails twice: $i(Q_{8 \times 11}) = 6 > 5 = i(Q_{9 \times 11}) = i(Q_{10 \times 11}) = i(Q_{11 \times 11})$, and $i(Q_{11 \times 18}) = 9 > 8 = i(Q_{12 \times 18})$.

Published
2016

3. 12108

Author

Pan, Yifei and Weakley, William D.

Published
2019

5. Problems

Author

Edwards, Steve, Whitenton, James, Huff, Barthel Wayne, Herschkorn, Stephen J., Bencze, Mihály, Weakley, William D., Klamkin, Murray S., Manea, Mihai, Zhou, Li, Bataille, Michel, Can, Minh, Christopher, John, Dale, Knut, Donini, Daniele, Foster, Lorraine L., Johnson, Leslie D., Lau, Kee-Wai, Seiffert, Heinz-Jürgen, Sinefakopoulos, Achilleas, Singer, Nicholas C., Bracken, Paul, Wenchang, Chu, Bhanu, K. S., Deshpande, M. N., Binz, Jany C., Bradie, Brian, Buelk, Samuel J., Byrd, Stan, Case, Jeremy, De Donno, Livia, Díaz-Barrero, José Luis, Getz, Marty, Jones, Dixon, Gonchigdanzan, Hurlee, Hysnelaj, Enkel, Kaczkowski, Stephen, Krummel, Brian, Luk, Lau Sai, Malm, Eric, Mansuri, Juniad N., Poodiack, Robert, Pratt, Rob, Roach, Nelisa, Rumula, Yisa, Richberg, Rolf, Saleh, Ossama A., Walters, Terry J., Simion, Andrei, Singh, Satyanand, Spellman, John W., Torrejón, Ricardo M., Spivey, Mike, Tang, H. T., Vowe, Michael, Klambin, Murray S., Barbara, Roy, Brown, Ethan S., Hillar, Christopher J., Curtin, Eugene, McCoy, T. L., Thornber, Nora, Wick, Darren D., Abad, J. P. V., Andreolli, Michael, Bowron, Mark, Brodie, Marc, Falkowski, B. J., Fleishman, Dimitry, Grossman, Jerrold W., Lewis, Kathleen E., Long, J. Andrew, Magidin, Arturo, Reyes, Manuel, Sealy, R. P., Zwier, Paul J., Curtis, Chip, Baker, John A., and Hill, Christopher

Published
2004
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6. Problems

Author

Deutsch, Emeric, Niculescu, Constantin P., Hoehn, Larry, Quet, Leroy, Just, Erwin, Klamkin, Murray S., Weakley, William D., Wardlaw, William P., Rubenstein, Yanir A., Sumner, John S., Dove, Kevin L., Barbara, Roy, Lee, Ho-joo, Tarabay, Ajaj A., Ghalayini, Bassam B., and Doucette, Robert L.

Published
2001
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7. Problems

Author

Weakley, William D., Rubinstein, Yanir A., Lee, Ho-joo, Klamkin, Murray S., Tudoran, Razvan, Rosenholtz, Ira, Vowe, Michael, Nieto, Jose H., Delany, Jim, Amrahov, Shahin, Getz, Marty, Jones, Dixon, Binz, J. C., Edgar, Gerald A., Ferland, Kevin, Luca, Florian, Bloom, David M., Deutsch, Emeric, and Zhou, Li

Published
2000
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10. Problems

Author

Café Dalat Problem Solving Group, Cal Poly Pomona Problem Solving Group, Hajja, Mowaffaq, Furdui, Ovidiu, ShahAli, H. A., Gilbert, George, Abu-Saymeh, Sadi, Kahan, Steven, Edwards, Steve, Whitenton, James, Huff, Barthel Wayne, Pratt, Rob, Herschkorn, Stephen J., Miller, Andrew J., Bencze, Mihály, Reid, Michael, Weakley, William D., and Delany, Jim

Published
2005
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18. Problems

Author

Hajja, Mowaffaq, primary, Furdui, Ovidiu, additional, ShahAli, H. A., additional, Gilbert, George, additional, Abu-Saymeh, Sadi, additional, Kahan, Steven, additional, Edwards, Steve, additional, Whitenton, James, additional, Huff, Barthel Wayne, additional, Pratt, Rob, additional, Herschkorn, Stephen J., additional, Miller, Andrew J., additional, Bencze, Mihály, additional, Reid, Michael, additional, Weakley, William D., additional, and Delany, Jim, additional

Published
2005
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35. Optimal binary covering codes of length 2j

Author

Weakley, William D.

Abstract

The minimum size of a binary covering code of length n and covering radius r is denoted by K(n,r), and codes of this length are called optimal. For j > 0 and n = 2j, it is known that K(n,1) = 2 · K(n−1,1) = 2n − j. Say that two binary words of length n form a duo if the Hamming distance between them is 1 or 2. In this paper, it is shown that each optimal binary covering code of length n = 2j, j > 0, and covering radius 1 is the union of duos in just one way, and that the closed neighborhoods of the duos form a tiling of the set of binary words of length n. Methods of constructing such optimal codes from optimal covering codes of length n − 1 (that is, perfect single‐error‐correcting codes) are discussed. The paper ends with the construction of an optimal covering code of length 16 that does not contain an extension of any optimal covering code of length 15. © 2005 Wiley Periodicals, Inc. J Combin Designs

Published
2006
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