{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T22:54:32Z","timestamp":1649199272007},"reference-count":26,"publisher":"World Scientific Pub Co Pte Lt","issue":"03n04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Inter. Net."],"published-print":{"date-parts":[[2016,9]]},"abstract":"<jats:p>A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted by<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\" altimg=\"eq-00001.gif\"><mml:mrow><mml:msub><mml:mrow><mml:mtext>la<\/mml:mtext><\/mml:mrow><mml:mi>k<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>, is the least number of linear k-forests needed to decompose G. Recently, Zuo, He, and Xue studied the exact values of the linear<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\" altimg=\"eq-00002.gif\"><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo>\u2212<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>-arboricity of Cartesian products of various combinations of complete graphs, cycles, complete multipartite graphs. In this paper, for general k we show that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\" altimg=\"eq-00003.gif\"><mml:mrow><mml:mi>max<\/mml:mi><mml:mo>{<\/mml:mo><mml:msub><mml:mrow><mml:mtext>la<\/mml:mtext><\/mml:mrow><mml:mi>k<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mtext>la<\/mml:mtext><\/mml:mrow><mml:mi>l<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>}<\/mml:mo><mml:mo>\u2264<\/mml:mo><mml:msub><mml:mrow><mml:mtext>la<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>max<\/mml:mi><mml:mo>{<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>l<\/mml:mi><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo>\u25a1<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2264<\/mml:mo><mml:msub><mml:mrow><mml:mtext>la<\/mml:mtext><\/mml:mrow><mml:mi>k<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:mtext>la<\/mml:mtext><\/mml:mrow><mml:mi>l<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>for any two graphs G and H. Denote by<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\" altimg=\"eq-00004.gif\"><mml:mrow><mml:mi>G<\/mml:mi><mml:mo>\u2218<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo>,<\/mml:mo><mml:mtext>\u2009<\/mml:mtext><mml:mi>G<\/mml:mi><mml:mo>\u00d7<\/mml:mo><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\" altimg=\"eq-00005.gif\"><mml:mrow><mml:mi>G<\/mml:mi><mml:mo>\u22a0<\/mml:mo><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>the lexicographic product, direct product and strong product of two graphs G and H, respectively. For any two graphs G and H, we also derive upper and lower bounds of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\" altimg=\"eq-00006.gif\"><mml:mrow><mml:msub><mml:mrow><mml:mtext>la<\/mml:mtext><\/mml:mrow><mml:mi>k<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo>\u2218<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mtext>la<\/mml:mtext><\/mml:mrow><mml:mi>k<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo>\u00d7<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\" altimg=\"eq-00007.gif\"><mml:mrow><mml:msub><mml:mrow><mml:mtext>la<\/mml:mtext><\/mml:mrow><mml:mi>k<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo>\u22a0<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>in this paper. The linear k-arboricity of a 2-dimensional grid graph, a r-dimensional mesh, a r-dimensional torus, a r-dimensional generalized hypercube and a hyper Petersen network are also studied.<\/jats:p>","DOI":"10.1142\/s0219265916500080","type":"journal-article","created":{"date-parts":[[2017,1,23]],"date-time":"2017-01-23T08:34:57Z","timestamp":1485160497000},"page":"1650008","source":"Crossref","is-referenced-by-count":0,"title":["Linear<i>k<\/i>-Arboricity in Product Networks"],"prefix":"10.1142","volume":"16","author":[{"given":"YAPING","family":"MAO","sequence":"first","affiliation":[{"name":"Department of Mathematics, Qinghai Normal University, Xining, Qinghai 810008, China"},{"name":"Key Laboratory of IOT of Qinghai Province, Xining, Qinghai 810008, China"}]},{"given":"ZHIWEI","family":"GUO","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Qinghai Normal University, Xining, Qinghai 810008, China"}]},{"given":"NAN","family":"JIA","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Qinghai Normal University, Xining, Qinghai 810008, China"}]},{"given":"HE","family":"LI","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Qinghai Normal University, Xining, Qinghai 810008, China"}]}],"member":"219","published-online":{"date-parts":[[2017,1,23]]},"reference":[{"key":"p_2","first-page":"307","volume":"17","author":"Habib B","year":"1983","journal-title":"Annales Polonici Mathematici"},{"key":"p_3","doi-asserted-by":"publisher","DOI":"10.1111\/j.1749-6632.1970.tb56470.x"},{"key":"p_4","first-page":"97","volume":"18","author":"Aldred N.C","year":"1998","journal-title":"Australasian J. 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