{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T01:40:05Z","timestamp":1750470005279,"version":"3.41.0"},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2007,5,1]],"date-time":"2007-05-01T00:00:00Z","timestamp":1177977600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2007,5]]},"abstract":"<jats:p>Let <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub> be graphs of order <jats:italic>n<\/jats:italic> with maximum degree \u0394<jats:sub>1<\/jats:sub> and \u0394<jats:sub>2<\/jats:sub>, respectively. <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub> are said to <jats:italic>pack<\/jats:italic> if there exist injective mappings of the vertex sets into [<jats:italic>n<\/jats:italic>], such that the images of the edge sets do not intersect. Sauer and Spencer showed that if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548306007929_inline1\">\n\t      <jats:alt-text>$\\Delta_1 \\Delta_2&lt;\\frac{n}{2}$<\/jats:alt-text>\n\t    <\/jats:inline-graphic>, then <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub> pack. We extend this result by showing that if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548306007929_inline1\">\n\t      <jats:alt-text>$\\Delta_1 \\Delta_2 \\leq \\frac{n}{2}$<\/jats:alt-text>\n\t    <\/jats:inline-graphic>, then <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub> do not pack if and only if one of <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> or <jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub> is a perfect matching and the other either is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548306007929_inline2\">\n\t      <jats:alt-text>$K_{\\frac{n}{2},\\frac{n}{2}}$<\/jats:alt-text>\n\t    <\/jats:inline-graphic> with <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548306007929_inline3\">\n\t      <jats:alt-text>$\\frac{n}{2}$<\/jats:alt-text>\n\t    <\/jats:inline-graphic> odd or contains <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548306007929_inline4\">\n\t      <jats:alt-text>$K_{\\frac{n}{2} + 1}$<\/jats:alt-text>\n\t    <\/jats:inline-graphic>.<\/jats:p>","DOI":"10.1017\/s0963548306007929","type":"journal-article","created":{"date-parts":[[2006,8,14]],"date-time":"2006-08-14T09:26:51Z","timestamp":1155547611000},"page":"409-416","source":"Crossref","is-referenced-by-count":8,"title":["Extremal Graphs for a Graph Packing Theorem of Sauer and Spencer"],"prefix":"10.1017","volume":"16","author":[{"given":"HEMANSHU","family":"KAUL","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ALEXANDR","family":"KOSTOCHKA","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2007,5,1]]},"reference":[{"key":"S0963548306007929_manual_ref-2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(78)90030-8"},{"key":"S0963548306007929_manual_ref-3","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(74)90119-8"},{"key":"S0963548306007929_manual_ref-4","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(78)90005-9"},{"volume-title":"Extremal Graph Theory","year":"1978","author":"Bollob\u00e1s","key":"S0963548306007929_manual_ref-1"},{"key":"S0963548306007929_manual_ref-5","first-page":"78","article-title":"Packing of graphs","volume":"362","author":"Wozniak","year":"1997","journal-title":"Dissertationes Math."},{"key":"S0963548306007929_manual_ref-6","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(88)90232-4"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548306007929","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T01:03:55Z","timestamp":1750467835000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548306007929\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,5]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["S0963548306007929"],"URL":"https:\/\/doi.org\/10.1017\/s0963548306007929","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2007,5]]}}}