{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:09:59Z","timestamp":1760058599684,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T00:00:00Z","timestamp":1744588800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Walailak University under the New Researcher Development scheme","award":["WU67234"],"award-info":[{"award-number":["WU67234"]}]},{"name":"Centre of Excellence in Mathematics, Ministry of Higher Education, Science, Research, and Innovation, Thailand","award":["WU67234"],"award-info":[{"award-number":["WU67234"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we show that if a planar graph G satisfies the following conditions: (i) none of its 3-faces is adjacent to a 6\u2212-face, and (ii) none of its 4-faces is adjacent to a 5\u2212-face, then V(G) can be partitioned into two subsets, each induces a forest, while one of the forests has maximum degree of at most 2. This result implies that every planar graph (i) with no chordal 7\u2212-cycles, or (ii) with neither 4- nor 6-cycles, also admits such a partition.<\/jats:p>","DOI":"10.3390\/axioms14040293","type":"journal-article","created":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T04:42:07Z","timestamp":1744605727000},"page":"293","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Partitioning Planar Graphs Without Specific Cycles into a Forest and a Disjoint Union of Paths"],"prefix":"10.3390","volume":"14","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\/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":[[2025,4,14]]},"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. Graph Theory"},{"key":"ref_2","first-page":"18","article-title":"A proof of Gr\u00fcnbaum\u2019s conjecture on the acyclic 5-colorability of planar graphs","volume":"231","author":"Borodin","year":"1976","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1002\/jgt.3190140108","article-title":"On the linear vertex-arboricity of a plane graph","volume":"14","author":"Poh","year":"1990","journal-title":"J. Graph Theory"},{"key":"ref_4","first-page":"109","article-title":"Ein Dreifarbensatz f\u00fcr dreikreisfreie Netze auf der Kugel","volume":"8","year":"1959","journal-title":"Math. Nat. Reihe"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.ejc.2017.06.014","article-title":"Partitioning a triangle-free planar graph into a forest and a forest of bounded degree","volume":"66","author":"Dross","year":"2017","journal-title":"Eur. J. Comb."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111577","DOI":"10.1016\/j.disc.2019.06.033","article-title":"Planar graphs with girth at least 5 are (3,4)-colorable","volume":"342","author":"Choi","year":"2019","journal-title":"Discret. Math."},{"key":"ref_7","first-page":"34","article-title":"On the partition of a planar graph of girth 5 into an empty and an acyclic subgraph","volume":"8","author":"Borodin","year":"2001","journal-title":"Diskret. Anal. Issled. Oper."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1016\/j.jctb.2008.11.002","article-title":"Decomposing a planar graph of girth 5 into an independent set and a forest","volume":"99","author":"Kawarabayashi","year":"2009","journal-title":"J. Combin. Theory S. B"},{"key":"ref_9","first-page":"125032","article-title":"On the vertex partition of planar graphs into forests with bounded degree","volume":"374","author":"Wang","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.jctb.2013.10.002","article-title":"Defective 2-coloring of sparse graphs","volume":"104","author":"Borodin","year":"2014","journal-title":"J. Combin. Theory Ser. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1002\/jgt.20155","article-title":"Improper choosability of graphs and maximum average degree","volume":"52","author":"Havet","year":"2006","journal-title":"J. Graph Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1002\/jgt.22734","article-title":"An (F1,F4)-partition of graphs with low genus and girth at least 6","volume":"99","author":"Chen","year":"2021","journal-title":"J. Graph Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/3-540-33700-8_21","article-title":"Threshold for path colorings of planar graphs","volume":"26","author":"Chappell","year":"2005","journal-title":"Algorithms Combin."},{"key":"ref_14","first-page":"1","article-title":"Partitioning sparse graphs into an independent set and a forest of bounded degree","volume":"25","author":"Dross","year":"2018","journal-title":"Electron. J. Comb."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1112\/jlms\/s1-44.1.612","article-title":"The point-arboricity of planar graphs","volume":"44","author":"Chartrand","year":"1969","journal-title":"J. Lond. Math. Soc."},{"key":"ref_16","first-page":"1","article-title":"Near-colorings: Non-colorable graphs and NP-completeness","volume":"22","author":"Montassier","year":"2015","journal-title":"Electron. J. Comb."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"103878","DOI":"10.1016\/j.ejc.2023.103878","article-title":"Decomposing a triangle-free planar graph into a forest and a subcubic forest","volume":"116","author":"Feghali","year":"2024","journal-title":"Eur. J. Combin."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2142","DOI":"10.1016\/j.disc.2018.04.024","article-title":"Defective 2-colorings of planar graphs without 4-cycles and 5-cycles","volume":"341","author":"Sittitrai","year":"2018","journal-title":"Discret. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"112172","DOI":"10.1016\/j.disc.2020.112172","article-title":"Partitioning planar graphs without 4-cycles and 5-cycles into bounded degree forests","volume":"344","author":"Cho","year":"2021","journal-title":"Discret. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1552","DOI":"10.1007\/s11425-007-0106-4","article-title":"Planar graphs without 4, 6, 8-cycles are 3-colorable","volume":"50","author":"Wang","year":"2007","journal-title":"Sci. China A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.disc.2015.08.023","article-title":"The 3-colorability of planar graphs without cycles of length 4, 6 and 9","volume":"339","author":"Kang","year":"2016","journal-title":"Discret. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1016\/j.dam.2020.04.017","article-title":"Planar graphs without short even cycles are near-bipartite","volume":"284","author":"Liu","year":"2020","journal-title":"Discrete Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.dam.2024.05.015","article-title":"Planar graphs without 4- and 6-cycles are (3,4)-colorable","volume":"356","author":"Nakprasit","year":"2024","journal-title":"Discrete Appl. Math."},{"key":"ref_24","unstructured":"Hu, K., and Huang, M. (2025, March 18). An (F3,F4)-Partition of Planar Graphs Without 4- and 6-Cycles. Available online: https:\/\/ssrn.com\/abstract=4959839."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1016\/j.ipl.2013.06.001","article-title":"Planar graphs with cycles of length neither 4 nor 6 are (2,0,0)-colorable","volume":"113","author":"Wang","year":"2013","journal-title":"Inf. Proc. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1007\/s00373-022-02605-9","article-title":"Partitioning planar graphs without 4-cycles and 6-cycles into a linear forest and a forest","volume":"39","author":"Huang","year":"2023","journal-title":"Graphs Combin."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s40840-023-01528-9","article-title":"A Weak DP-Partitioning of planar graphs without 4-cycles and 6-cycles","volume":"46","author":"Sittitrai","year":"2023","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.dam.2022.04.020","article-title":"A sufficient condition for a planar graph to be (F,F2)-partitionable","volume":"318","author":"Liu","year":"2022","journal-title":"Discret. Appl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"112758","DOI":"10.1016\/j.disc.2021.112758","article-title":"(1,0,0)-colorability of planar graphs without cycles of length 4 or 6","volume":"345","author":"Kang","year":"2022","journal-title":"Discret. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/293\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:14:04Z","timestamp":1760030044000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/293"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,14]]},"references-count":29,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040293"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040293","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,4,14]]}}}