{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:49:29Z","timestamp":1760150969810,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,31]],"date-time":"2022-01-31T00:00:00Z","timestamp":1643587200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSRF via the Program Management Unit for Human Resources  Institutional Development, Research and Innovation","award":["B05F640106"],"award-info":[{"award-number":["B05F640106"]}]},{"name":"Research Fund for Supporting Lecturer to Admit High Potential Student to Study and Research on His Expert Program Year 2021 from the Graduate School Khon Kaen University, Thailand","award":["641T104-C"],"award-info":[{"award-number":["641T104-C"]}]},{"DOI":"10.13039\/501100004704","name":"National Research Council of Thailand","doi-asserted-by":"publisher","award":["N41A640141"],"award-info":[{"award-number":["N41A640141"]}],"id":[{"id":"10.13039\/501100004704","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color classes differ by at most one. An equitable (q,r)-tree-coloring of a graph G is an equitable q-coloring of G such that the subgraph induced by each color class is a forest of maximum degree at most r. Let the strong equitable vertex r-arboricity of a graph G, denoted by var\u2261(G), be the minimum p such that G has an equitable (q,r)-tree-coloring for every q\u2265p. The values of va1\u2261(Kn,n) were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of va1\u2261(Kn,n) were found in some special cases. In this paper, we extend their results by giving the exact values of va1\u2261(Kn,n) for all cases. In the process, we introduce a new function related to an equitable coloring and obtain a more general result by determining the exact value of each va1\u2261(Km,n) and va1\u2261(G) where G is a balanced complete k-partite graph Kn,\u2026,n. Both complete bipartite graphs Km,n and balanced complete k-partite graphs Kn,\u2026,n are symmetry in several aspects and also studied broadly. For the other aspect of symmetry, by the definition of equitable k-coloring of graphs, in a specific case that the number of colors divides the number of vertices of graph, we can say that the graph is a balanced k-partite graph.<\/jats:p>","DOI":"10.3390\/sym14020287","type":"journal-article","created":{"date-parts":[[2022,1,31]],"date-time":"2022-01-31T08:20:29Z","timestamp":1643617229000},"page":"287","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Strong Equitable Vertex 1-Arboricity of Complete Bipartite Graphs and Balanced Complete k-Partite Graphs"],"prefix":"10.3390","volume":"14","author":[{"given":"Janejira","family":"Laomala","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]},{"given":"Keaitsuda Maneeruk","family":"Nakprasit","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]},{"given":"Kittikorn","family":"Nakprasit","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1564-2642","authenticated-orcid":false,"given":"Watcharintorn","family":"Ruksasakchai","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Statistics and Computer Science, Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng Saen Campus, Nakhon Pathom 73140, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"920","DOI":"10.1080\/00029890.1973.11993408","article-title":"Equitable Coloring","volume":"80","author":"Meyer","year":"1973","journal-title":"Am. 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Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/287\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:11:45Z","timestamp":1760134305000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/287"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,31]]},"references-count":11,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["sym14020287"],"URL":"https:\/\/doi.org\/10.3390\/sym14020287","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,31]]}}}