{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T21:10:01Z","timestamp":1723065001421},"reference-count":0,"publisher":"Wiley","issue":"61","license":[{"start":{"date-parts":[[2000,1,1]],"date-time":"2000-01-01T00:00:00Z","timestamp":946684800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>This note explores a new family of graphs defined on the set of paths of the <jats:italic>m<\/jats:italic> \u00d7 <jats:italic>n<\/jats:italic> lattice. We let each of the paths of the lattice be represented by a vertex, and connect two vertices by an edge if the corresponding paths share more than <jats:italic>k<\/jats:italic>\nsteps, where <jats:italic>k<\/jats:italic>\nis a fixed parameter 0 = <jats:italic>k<\/jats:italic> = <jats:italic>m<\/jats:italic> + <jats:italic>n<\/jats:italic>. Each such graph is denoted by <jats:italic>G<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>n<\/jats:italic>, <jats:italic>k<\/jats:italic>). Two large complete subgraphs of <jats:italic>G<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>n<\/jats:italic>, <jats:italic>k<\/jats:italic>)\nare described for all values of <jats:italic>m<\/jats:italic>, <jats:italic>n<\/jats:italic>, and <jats:italic>k<\/jats:italic>. The size of the edge set is determined for <jats:italic>n<\/jats:italic> = 2, and a complicated recursive formula is given for the size of the edge set when <jats:italic>k<\/jats:italic> = 1.<\/jats:p>","DOI":"10.1155\/s0161171204306058","type":"journal-article","created":{"date-parts":[[2004,11,28]],"date-time":"2004-11-28T14:30:55Z","timestamp":1101652255000},"page":"3291-3299","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the edge set of graphs of lattice paths"],"prefix":"10.1155","volume":"2004","author":[{"given":"Steven","family":"Klee","sequence":"first","affiliation":[]},{"given":"Lara","family":"Pudwell","sequence":"additional","affiliation":[]},{"given":"Rick","family":"Gillman","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2004,11,28]]},"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2004\/709624.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/S0161171204306058","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T20:45:04Z","timestamp":1723063504000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/S0161171204306058"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":0,"journal-issue":{"issue":"61","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1155\/S0161171204306058"],"URL":"https:\/\/doi.org\/10.1155\/s0161171204306058","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2004,1]]},"assertion":[{"value":"2003-06-04","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2004-11-28","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}