GDD with equal number of even and odd blocks

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• 作者：Issa Ndungo^a ; Dinesh G. Sarvate^b ; ^{sarvated@cofc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Group divisible designs ; Latin squares ; Even and odd GDDs ; IMOLS ; SOLS
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 April 2016
• 年：2016
• 卷：339
• 期：4
• 页码：1344-1354
• 全文大小：421 K

文摘

We prove that the necessary condition, an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si2.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=5f82614ed52989c1ffd3fb69b2d7ad09" title="Click to view the MathML source">n≡0an>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">n≡0ath>an>an>an>(mod 3), is sufficient for the existence of GDD(an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si3.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=2aaec2d0a9e89729adcde8930aa555ca" title="Click to view the MathML source">n,2,4;3,4an>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">n,2,4;3,4ath>an>an>an>) except possibly for an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si4.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=f0693ef022d16094eb214d30d0d383e5" title="Click to view the MathML source">n=18an>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">n=18ath>an>an>an>. We prove that necessary conditions for the existence of group divisible designs GDD(an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si5.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=a7052da2db933112574daf25e3d02e97" title="Click to view the MathML source">n,2,4;λ₁,λ₂an>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">n,2,4;λ1,λ2ath>an>an>an>) with equal number of even and odd blocks are sufficient for GDDan id="mmlsi6" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si6.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=d937f820643441bd9ff6200954ed57da" title="Click to view the MathML source">(n,2,4;5n,7(n−1))an>an class="mathContainer hidden">an class="mathCode">ath altimg="si6.gif" overflow="scroll">(n,2,4;5n,7(n−1))ath>an>an>an> for all an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si7.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=c25734011ae51ca8c090609a5b2e6c35" title="Click to view the MathML source">n≥2an>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">n≥2ath>an>an>an>, GDDan id="mmlsi8" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si8.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=dd3009d18700f2bd92b11307fc2211e2" title="Click to view the MathML source">(7s,2,4;5s,7s−1)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si8.gif" overflow="scroll">(7s,2,4;5s,7s−1)ath>an>an>an> for all an id="mmlsi9" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si9.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=ba6f6c83975a5243fbb9c0409973a951" title="Click to view the MathML source">san>an class="mathContainer hidden">an class="mathCode">ath altimg="si9.gif" overflow="scroll">sath>an>an>an>, GDDan id="mmlsi10" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si10.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=2a4caf447e19588abe68a98980053e47" title="Click to view the MathML source">(5t+1,2,4;5t+1,7t)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">(5t+1,2,4;5t+1,7t)ath>an>an>an> for an id="mmlsi11" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si11.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=afd44a75e15363de04af8c8ad291c7ea" title="Click to view the MathML source">t≡an>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">t≡ath>an>an>an> 0(mod 2) and GDDan id="mmlsi12" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si12.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=baf84dbb6d67ee57b13d1c150636f79a" title="Click to view the MathML source">(5t+1,2,4;2(5t+1),14t)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si12.gif" overflow="scroll">(5t+1,2,4;2(5t+1),14t)ath>an>an>an> for all an id="mmlsi13" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si13.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=0ef6b8279b47aa1337ef089c00d67002" title="Click to view the MathML source">tan>an class="mathContainer hidden">an class="mathCode">ath altimg="si13.gif" overflow="scroll">tath>an>an>an>. To complete the existence of such GDDs, one needs to construct two more families: GDDan id="mmlsi10" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si10.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=2a4caf447e19588abe68a98980053e47" title="Click to view the MathML source">(5t+1,2,4;5t+1,7t)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">(5t+1,2,4;5t+1,7t)ath>an>an>an> for all odd an id="mmlsi13" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si13.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=0ef6b8279b47aa1337ef089c00d67002" title="Click to view the MathML source">tan>an class="mathContainer hidden">an class="mathCode">ath altimg="si13.gif" overflow="scroll">tath>an>an>an>, and GDDan id="mmlsi16" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si16.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=79ec5f2063722d8ad2a9d68096c95ace" title="Click to view the MathML source">(35s+21,2,4;5s+3,7s+4)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si16.gif" overflow="scroll">(35s+21,2,4;5s+3,7s+4)ath>an>an>an> for all positive integers an id="mmlsi9" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15004033&_mathId=si9.gif&_user=111111111&_pii=S0012365X15004033&_rdoc=1&_issn=0012365X&md5=ba6f6c83975a5243fbb9c0409973a951" title="Click to view the MathML source">san>an class="mathContainer hidden">an class="mathCode">ath altimg="si9.gif" overflow="scroll">sath>an>an>an>.