{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,21]],"date-time":"2024-06-21T14:09:08Z","timestamp":1718978948517},"reference-count":11,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,5]],"date-time":"2006-10-05T00:00:00Z","timestamp":1160006400000},"content-version":"vor","delay-in-days":6243,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1989,9]]},"abstract":"Abstract<\/jats:title>Let \u03b8 be a family of graphs. By a \u03b8\u2010decomposition of a graph G<\/jats:italic> we mean a partition \u03bb of the edge set of G<\/jats:italic> such that every F<\/jats:italic> \u03f5 \u03c0 spans in G<\/jats:italic> a subgraph isomorphic to a graph in \u03b8.<\/jats:p>In this paper we state the following conjecture: If T<\/jats:italic>1<\/jats:sub> and T2<\/jats:sub> are two trees having relatively prime sizes then there exists c<\/jats:italic> = c(T1<\/jats:sub> T2<\/jats:sub>) such that every graph G<\/jats:italic> satisfying the condition \u03b4(G) \u2a7e c has a {T<\/jats:italic>1<\/jats:sub>, T2<\/jats:sub>}\u2010decom\u2010position. We prove this conjecture for some special pairs of trees. In particular, we prove it in the following cases:<\/jats:p>(i) T<\/jats:italic>1<\/jats:sub> and T2<\/jats:sub><\/jats:italic> are stars having relatively prime sizes;<\/jats:p>(ii) T1<\/jats:sub><\/jats:italic> and T2<\/jats:sub><\/jats:italic> are paths having relatively prime sizes; and.<\/jats:p>(iii) T1<\/jats:sub><\/jats:italic> = T2<\/jats:sub><\/jats:italic> \u2010 {v}, where v is a terminal vertex in T<\/jats:bold><\/jats:italic>2<\/jats:sub>.<\/jats:p>","DOI":"10.1002\/jgt.3190130403","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T12:26:14Z","timestamp":1180182374000},"page":"393-403","source":"Crossref","is-referenced-by-count":1,"title":["Decompositions of graphs into trees"],"prefix":"10.1002","volume":"13","author":[{"given":"Zbigniew","family":"Lonc","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,5]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01956769"},{"key":"e_1_2_1_3_2","unstructured":"M.Behzad G.Chartrand andL.Foster\u2010Lesniak Graphs and Digraphs. Wadsworth International Group Belmont (1979)."},{"key":"e_1_2_1_4_2","unstructured":"J. C.BermondandD.Sotteau Graph decompositions andG\u2010designs.Proceedings of the Fifth British Combinatorial Conference Aberdeen (1975)53\u201372."},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-349-03521-2"},{"key":"e_1_2_1_6_2","first-page":"29","article-title":"Chromatic decomposition of a graph","volume":"21","author":"Caro Y.","year":"1986","journal-title":"Ars. Combinat."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511662157.007"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364"},{"key":"e_1_2_1_9_2","first-page":"295","article-title":"Decomposition of graphs into graphs with bounded maximum degrees","volume":"18","author":"Lonc Z.","year":"1985","journal-title":"Demonstratio Math."},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00582740"},{"key":"e_1_2_1_11_2","volume-title":"Combinatorial Problems and Exercises","author":"Lova\u00e1sz L.","year":"1979"},{"key":"e_1_2_1_12_2","first-page":"201","article-title":"Decompositions of graphs into forests of equal size and with bounded maximum degrees","volume":"21","author":"Truszczy\u0144ski M.","year":"1986","journal-title":"Ars Combinat."}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190130403","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190130403","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T14:03:09Z","timestamp":1697983389000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190130403"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,9]]},"references-count":11,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1989,9]]}},"alternative-id":["10.1002\/jgt.3190130403"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190130403","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,9]]}}}