{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T06:07:04Z","timestamp":1648966024105},"reference-count":15,"publisher":"World Scientific Pub Co Pte Lt","issue":"01n02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Inter. Net."],"published-print":{"date-parts":[[2010,3]]},"abstract":"<jats:p> Assume that n is a positive integer with n \u2265 4 and F is a subset of the edges of the hypercube Q<jats:sub>n<\/jats:sub> with |F| \u2264 n-4. Let u , x be two distinct white vertices of Q<jats:sub>n<\/jats:sub> and v , y be two distinct black vertices of Q<jats:sub>n<\/jats:sub>, where black and white refer to the two parts of the bipartition of Q<jats:sub>n<\/jats:sub>. Let l<jats:sub>1<\/jats:sub> and l<jats:sub>2<\/jats:sub> be odd integers, where l<jats:sub>1<\/jats:sub> \u2265 d<jats:sub>Q<jats:sub>n-F<\/jats:sub><\/jats:sub>( u , v ), l<jats:sub>2<\/jats:sub> \u2265 d<jats:sub>Q<jats:sub>n-F<\/jats:sub><\/jats:sub>( x , y ), and l<jats:sub>1<\/jats:sub> + l<jats:sub>2<\/jats:sub> = 2<jats:sup>n<\/jats:sup> - 2. Moreover, let l<jats:sub>3<\/jats:sub> and l<jats:sub>4<\/jats:sub> be even integers, where l<jats:sub>3<\/jats:sub> \u2265 d<jats:sub>Q<jats:sub>n-F<\/jats:sub><\/jats:sub>( u , x ), l<jats:sub>4<\/jats:sub> \u2265 d<jats:sub>Q<jats:sub>n-F<\/jats:sub><\/jats:sub>( v , y ), and l<jats:sub>3<\/jats:sub>+l<jats:sub>4<\/jats:sub> = 2<jats:sup>n<\/jats:sup> - 2. In this paper, we prove that there are two disjoint paths P<jats:sub>1<\/jats:sub> and P<jats:sub>2<\/jats:sub> such that (1) P<jats:sub>1<\/jats:sub> is a path joining u to v with length l(P<jats:sub>1<\/jats:sub>) = l<jats:sub>1<\/jats:sub>, (2) P<jats:sub>2<\/jats:sub> is a path joining x to y with l(P<jats:sub>2<\/jats:sub>) = l<jats:sub>2<\/jats:sub>, and (3) P<jats:sub>1<\/jats:sub> \u222a P<jats:sub>2<\/jats:sub> spans Q<jats:sub>n<\/jats:sub> - F. Moreover, there are two disjoint paths P<jats:sub>3<\/jats:sub> and P<jats:sub>4<\/jats:sub> such that (1) P<jats:sub>3<\/jats:sub> is a path joining u to x with l(P<jats:sub>3<\/jats:sub>) = l<jats:sub>3<\/jats:sub>, (2) P<jats:sub>4<\/jats:sub> is a path joining v to y with l(P<jats:sub>4<\/jats:sub>) = l<jats:sub>4<\/jats:sub>, and (3) P<jats:sub>3<\/jats:sub> \u222a P<jats:sub>4<\/jats:sub> spans Q<jats:sub>n<\/jats:sub> - F except the following cases: (a) l<jats:sub>3<\/jats:sub> = 2 with d<jats:sub>Q<jats:sub>n-F<\/jats:sub><\/jats:sub>( u , x ) = 2 and d<jats:sub>Q<jats:sub>n-F-{ v , y }<\/jats:sub><\/jats:sub>( u , x ) &gt; 2, and (b) l<jats:sub>4<\/jats:sub> = 2 with d<jats:sub>Q<jats:sub>n-F<\/jats:sub><\/jats:sub>( v , y ) = 2 and d<jats:sub>Q<jats:sub>n-F-{ u , x }<\/jats:sub><\/jats:sub>( v , y ) &gt; 2. <\/jats:p>","DOI":"10.1142\/s021926591000274x","type":"journal-article","created":{"date-parts":[[2010,11,11]],"date-time":"2010-11-11T10:20:58Z","timestamp":1289470858000},"page":"71-96","source":"Crossref","is-referenced-by-count":1,"title":["ON THE EXISTENCE OF DISJOINT SPANNING PATHS IN FAULTY HYPERCUBES"],"prefix":"10.1142","volume":"11","author":[{"given":"CHENG-KUAN","family":"LIN","sequence":"first","affiliation":[{"name":"Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30010, R.O.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JIMMY J. M.","family":"TAN","sequence":"additional","affiliation":[{"name":"Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30010, R.O.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"LIH-HSING","family":"HSU","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Information Engineering, Providence University, Taichung, Taiwan 43301, R.O.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"EDDIE","family":"CHENG","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, U.S.A."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L\u00c1SZL\u00d3","family":"LIPT\u00c1K","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, U.S.A."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","volume-title":"Graph Theory with Applications","author":"Bondy J. 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