Regular sparse anti-magic squares with small odd densities

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• 作者：Guangzhou Chen^a ; Haiyan Chen^b ; Kejun Chen^c ; ^{yctuckj@163.com" class="auth_mail" title="E-mail the corresponding author} ; Wen Li^b
• 关键词：Magic squares ; Sparse anti-magic squares ; Regular ; Vertex-magic labeling
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 January 2016
• 年：2016
• 卷：339
• 期：1
• 页码：138-156
• 全文大小：408 K

文摘

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si1.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=bc68d04b1f4862cbc4f6887913a729b5" title="Click to view the MathML source">n×ndden">de">$n\times n$ array based on d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si2.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=1947bafe71402fb6170f2a920c83fb82" title="Click to view the MathML source">{0,1,…,nd}dden">de">$\{0,1,\dots ,n\mathrm{<font\; color="red">d</font>}\}$ is called a sparse anti-magic square of order  d="mmlsi3" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si3.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=b145e06f73531629ee6ca6a4b95af865" title="Click to view the MathML source">ndden">de">$n$with density  d="mmlsi4" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si4.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=c1fb50f914f11750cac0a48323737aa0" title="Click to view the MathML source">ddden">de">$\mathrm{<font\; color="red">d</font>}$ (d="mmlsi5" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si5.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=32972495fd44e7e17896946cd8a861fa" title="Click to view the MathML source">d<ndden">de">$\mathrm{<font\; color="red">d</font>}<n$), denoted by SAMSd="mmlsi6" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si6.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=14700298b67feb22eaa3292addd15738" title="Click to view the MathML source">(n,d)dden">de">$(n,\mathrm{<font\; color="red">d</font>})$, if its row-sums, column-sums and two main diagonal sums constitute a set of d="mmlsi7" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si7.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=362a992187dc1e51bbac2f0a52ddc3ca" title="Click to view the MathML source">2n+2dden">de">$2n+2$ consecutive integers. A SAMSd="mmlsi6" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si6.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=14700298b67feb22eaa3292addd15738" title="Click to view the MathML source">(n,d)dden">de">$(n,\mathrm{<font\; color="red">d</font>})$ is called regular   if there are d="mmlsi4" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si4.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=c1fb50f914f11750cac0a48323737aa0" title="Click to view the MathML source">ddden">de">$\mathrm{<font\; color="red">d</font>}$ positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares with densities d="mmlsi10" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si10.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=dc2987a33379a332178409e39252061f" title="Click to view the MathML source">d=3,5dden">de">$\mathrm{<font\; color="red">d</font>}=3,5$ and it is proved that there exists a regular SAMSd="mmlsi11" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si11.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=2323f696c8722edb5d0cfd690ddb00f7" title="Click to view the MathML source">(n,3)dden">de">$(n,3)$ if and only if d="mmlsi12" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si12.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=32bdb79f9d9eb84c72e627d1451a46fa" title="Click to view the MathML source">n≥4dden">de">$n\ge 4$ and there exists a regular SAMSd="mmlsi13" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si13.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=63a7e0519b6cdfc6455e5bb1f1289ea5" title="Click to view the MathML source">(n,5)dden">de">$(n,5)$ if and only if d="mmlsi14" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002939&_mathId=si14.gif&_user=111111111&_pii=S0012365X15002939&_rdoc=1&_issn=0012365X&md5=26cef2ff5f796d653b7ecb149be4365d" title="Click to view the MathML source">n≥6dden">de">$n\ge 6$.