Maximum packing for perfect four-triple configurations
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The graph consisting of the four 3-cycles (triples) style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml45&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=791635a5fd5cd118041be3ff41cffc5e"" title=""Click to view the MathML source"">(x<sub>1sub>,x<sub>2sub>,x<sub>8sub>),(x<sub>2sub>,x<sub>3sub>,x<sub>4sub>),(x<sub>4sub>,x<sub>5sub>,x<sub>6sub>), and style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml46&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=39881890b2e45c493cc4878a0c7cb553"" title=""Click to view the MathML source"">(x<sub>6sub>,x<sub>7sub>,x<sub>8sub>), where style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml47&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=bd8e32579576cd435b9dbe3fea0198b5"" title=""Click to view the MathML source"">x<sub>isub>'s are distinct, is called a 4-cycle-triple block and the 4-cycle style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml48&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=93899aa8cc48ff61c0c50008b1edb066"" title=""Click to view the MathML source"">(x<sub>2sub>,x<sub>4sub>,x<sub>6sub>,x<sub>8sub>) of the 4-cycle-triple block is called the interior (inside) 4-cycle. The graph consisting of the four 3-cycles style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml49&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=65f29ddfe0c66d71fd3ddfb712a864b2"" title=""Click to view the MathML source"">(x<sub>1sub>,x<sub>2sub>,x<sub>6sub>),(x<sub>2sub>,x<sub>3sub>,x<sub>4sub>),(x<sub>4sub>,x<sub>5sub>,x<sub>6sub>), and style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml50&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=bd2af6d06226c618852ead0d476590a3"" title=""Click to view the MathML source"">(x<sub>6sub>,x<sub>7sub>,x<sub>8sub>), where style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml51&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=f1834845b65c44374ad378680588ff87"" title=""Click to view the MathML source"">x<sub>isub>'s are distinct, is called a kite-triple block and the kite style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml52&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=a925c5ec8e9abe775acfb90fc64269de"" title=""Click to view the MathML source"">(x<sub>2sub>,x<sub>4sub>,x<sub>6sub>)-style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml53&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=2a8617715ae43573f265de19f7ec811a"" title=""Click to view the MathML source"">x<sub>8sub> (formed by a 3-cycle with a pendant edge) is called the interior kite. A decomposition of style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml54&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=d6f3931c4a1418aa99cfb634a1081f2b"" title=""Click to view the MathML source"">3kK<sub>nsub> into 4-cycle-triple blocks (or into kite-triple blocks) is said to be perfect if the interior 4-cycles (or kites) form a k-fold 4-cycle system (or kite system). A packing of style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml55&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=33162f7765043c0f9f07378b3df220fc"" title=""Click to view the MathML source"">3kK<sub>nsub> with 4-cycle-triples (or kite-triples) is a triple style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml56&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=d874aee78a751d9362245bb1cc8d37a1"" title=""Click to view the MathML source"">(X,B,L), where X is the vertex set of style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml57&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=7ac811711b2ef1e880a0756dce742b28"" title=""Click to view the MathML source"">K<sub>nsub>, B is a collection of 4-cycle-triples (or kite-triples), and L is a collection of 3-cycles, such that style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml58&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=26bf7a718c657ee71f7755bd57e5eb88"" title=""Click to view the MathML source"">Bsrc=""http://www.sciencedirect.com/scidirimg/entities/222a.gif"" alt=""union or logical sum"" border=0>L partitions the edge set of style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml59&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=0cac0d6950797da351cbdef42a6e85ec"" title=""Click to view the MathML source"">3kK<sub>nsub>. If style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml60&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=54659cf7599ffc786724b3e5c0861f7d"" title=""Click to view the MathML source"">|L| is as small as possible, or equivalently style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml61&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=1e6fdbde62ecde120af21b1df0a4bd56"" title=""Click to view the MathML source"">|B| is as large as possible, then the packing style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml62&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=d418e23d1d7c8c5216232f149772423d"" title=""Click to view the MathML source"">(X,B,L) is called maximum. If the maximum packing style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml63&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=9d4bf2398fab6c0761d3543b5807a10b"" title=""Click to view the MathML source"">(X,B,L) with 4-cycle-triples (or kite-triples) has the additional property that the interior 4-cycles (or kites) plus a specified subgraph of the leave L form a maximum packing of style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml64&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=dc15826ce6f722010015aefe335751ce"" title=""Click to view the MathML source"">kK<sub>nsub> with 4-cycles (or kites), it is said to be perfect.

This paper gives a complete solution to the problem of constructing perfect maximum packings of style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml65&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=2f92f38bc02343e56410a794c086dc61"" title=""Click to view the MathML source"">3kK<sub>nsub> with 4-cycle-triples and kite-triples, whenever n is the order of a style=""text-decoration:none; color:black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-4PF0XF6-3&_mathId=mml66&_user=10&_cdi=5632&_rdoc=14&_acct=C000050221&_version=1&_userid=10&md5=e125019090098012fe052098fe26fb80"" title=""Click to view the MathML source"">3k-fold triple system.

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