{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T05:32:46Z","timestamp":1762407166203,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T00:00:00Z","timestamp":1762214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010034","name":"Walailak University","doi-asserted-by":"crossref","award":["WU68268"],"award-info":[{"award-number":["WU68268"]}],"id":[{"id":"10.13039\/501100010034","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Centre of Excellence in Mathematics, Ministry of Higher Education, Science, Research, and Innovation, Thailand"},{"DOI":"10.13039\/501100004704","name":"National Research Council of Thailand","doi-asserted-by":"crossref","award":["N42A680154"],"award-info":[{"award-number":["N42A680154"]}],"id":[{"id":"10.13039\/501100004704","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>It has been shown that there is a planar graph without 3-cycles which is not (\u03941,\u03942)-colorable for any given \u03941,\u03942. This inspires many research to obtain sufficient conditions for planar graphs without 4-cycles and other cycles to be (\u03941,\u03942)-colorable. For example, planar graphs without k-cycles and 4-cycles are (\u03941,\u03942)-colorable for each k\u2208{3,5} and (\u03941,\u03942)={(2,6),(3,4)}. In this work, we study the values of \u03941 and \u03942 that make a planar graph without only 4-cycles (\u03941,\u03942)-colorable. We begin with \u03941=6 and \u03942=6. Planar graphs without 4-cycles are (6,6)-colorable. Two colors in a (\u03941,\u03942)-coloring, where \u03941=\u03942 are switchable, thus reflect the symmetry of the resulting coloring. Furthermore, some proof techniques are using the symmetry of these two colors.<\/jats:p>","DOI":"10.3390\/sym17111865","type":"journal-article","created":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T12:13:08Z","timestamp":1762258388000},"page":"1865","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Planar Graphs Without 4-Cycles Are (6, 6)-Colorable"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1770-8369","authenticated-orcid":false,"given":"Pongpat","family":"Sittitrai","sequence":"first","affiliation":[{"name":"Futuristic Science Research Center, School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand"},{"name":"Research Center for Theoretical Simulation and Applied Research in Bioscience and Sensing, Walailak University, Nakhon Si Thammarat 80160, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-2288-4920","authenticated-orcid":false,"given":"Wannapol","family":"Pimpasalee","sequence":"additional","affiliation":[{"name":"Department of Science and Mathematics, Faculty of Science and Health Technology, Kalasin University, Kalasin 46000, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7550-7744","authenticated-orcid":false,"given":"Keaitsuda Maneeruk","family":"Nakprasit","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"},{"name":"Centre of Excellence in Mathematics, Ministry of Higher Education, Science, Research and Innovation, Bangkok 10400, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0421-3631","authenticated-orcid":false,"given":"Kittikorn","family":"Nakprasit","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"},{"name":"Centre of Excellence in Mathematics, Ministry of Higher Education, Science, Research and Innovation, Bangkok 10400, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1002\/jgt.3190100207","article-title":"Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency","volume":"10","author":"Cowen","year":"1986","journal-title":"J. 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