On the existence of unparalleled even cycle systems

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• 作者：Peter Danziger ^{danziger@ryerson.ca" class="auth_mail" title="E-mail the corresponding author} ; Eric Mendelsohn ^{mendelso@math.utoronto.ca" class="auth_mail" title="E-mail the corresponding author} ; Tommaso Traetta ^{tommaso.traetta@ryerson.ca" class="auth_mail" title="E-mail the corresponding author traetta.tommaso@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：59
• 期：Complete
• 页码：11-22
• 全文大小：608 K

文摘

A class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si1.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=435330ac68568b7632288d7b885e1339" title="Click to view the MathML source">2tclass="mathContainer hidden">class="mathCode">$2t$-cycle system of even order class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si2.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=9e4a2e52dd96fb4dcc03172c2b5e3a2b" title="Click to view the MathML source">vclass="mathContainer hidden">class="mathCode">$v$ is a set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si3.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=1ae8d49cb65832f6adb48929b86efe13" title="Click to view the MathML source">Cclass="mathContainer hidden">class="mathCode">$\mathcal{C}$ of cycles of length class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si1.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=435330ac68568b7632288d7b885e1339" title="Click to view the MathML source">2tclass="mathContainer hidden">class="mathCode">$2t$ whose edges partition the edge-set of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si5.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=28560d27645a4aca952616aea4104f27" title="Click to view the MathML source">K_v−Iclass="mathContainer hidden">class="mathCode">$K_v-I$ (i.e., the complete graph minus the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si6.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=bbbf2ca9e1c4b321dfc5248ad8f8e0fd" title="Click to view the MathML source">1class="mathContainer hidden">class="mathCode">$1$-factor class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si7.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=ec4c77a82e00b27b76a3c8875e84ff7b" title="Click to view the MathML source">Iclass="mathContainer hidden">class="mathCode">$I$). If class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si8.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=c3db5586baa0a7d6848b2e61b94417cc">class="imgLazyJSB inlineImage" height="15" width="106" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S019566981630049X-si8.gif">class="mathContainer hidden">class="mathCode">$v\equiv 0(mod2t)$, a set of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si9.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=0f5eb5fc6cd5852e690bd6f3590de340" title="Click to view the MathML source">v/2tclass="mathContainer hidden">class="mathCode">$v/2t$ vertex-disjoint cycles of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si3.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=1ae8d49cb65832f6adb48929b86efe13" title="Click to view the MathML source">Cclass="mathContainer hidden">class="mathCode">$\mathcal{C}$ is a parallel class. If class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si3.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=1ae8d49cb65832f6adb48929b86efe13" title="Click to view the MathML source">Cclass="mathContainer hidden">class="mathCode">$\mathcal{C}$ has no parallel classes, we call such a system unparalleled.

We show that there exists an unparalleled class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si1.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=435330ac68568b7632288d7b885e1339" title="Click to view the MathML source">2tclass="mathContainer hidden">class="mathCode">$2t$-cycle system of order class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si8.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=c3db5586baa0a7d6848b2e61b94417cc">class="imgLazyJSB inlineImage" height="15" width="106" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S019566981630049X-si8.gif">class="mathContainer hidden">class="mathCode">$v\equiv 0(mod2t)$ if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981630049X&_mathId=si14.gif&_user=111111111&_pii=S019566981630049X&_rdoc=1&_issn=01956698&md5=3de3482a4698b694b7a58196aaecc1bd" title="Click to view the MathML source">v>2t>2class="mathContainer hidden">class="mathCode">$v>2t>2$.