{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T22:47:42Z","timestamp":1761864462627,"version":"build-2065373602"},"reference-count":7,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,4,26]],"date-time":"2025-04-26T00:00:00Z","timestamp":1745625600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,4,26]],"date-time":"2025-04-26T00:00:00Z","timestamp":1745625600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100006919","name":"Massachusetts Institute of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100006919","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Theory Comput Syst"],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>The 2-colorable perfect matching problem asks whether a graph can be colored with two colors so that each node has exactly one neighbor with the same color as itself. We prove that this problem is NP-complete, even when restricted to 2-connected 3-regular planar graphs. In 1978, Schaefer proved that this problem is NP-complete in general graphs, and claimed without proof that the same result holds when restricted to 3-regular planar graphs. Thus we fill in the missing proof of this claim, while simultaneously strengthening to 2-connected graphs (which implies existence of a perfect matching). We also prove NP-completeness of <jats:italic>k<\/jats:italic>-colorable perfect matching, for any fixed <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$k \\ge 2$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00224-025-10221-2","type":"journal-article","created":{"date-parts":[[2025,4,26]],"date-time":"2025-04-26T09:54:09Z","timestamp":1745661249000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["2-Colorable Perfect Matching is NP-complete in 2-Connected 3-Regular Planar Graphs"],"prefix":"10.1007","volume":"69","author":[{"given":"Erik D.","family":"Demaine","sequence":"first","affiliation":[]},{"given":"Kritkorn","family":"Karntikoon","sequence":"additional","affiliation":[]},{"given":"Nipun","family":"Pitimanaaree","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,26]]},"reference":[{"key":"10221_CR1","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1006\/jagm.2000.1132","volume":"38","author":"TC Biedl","year":"2001","unstructured":"Biedl, T.C., Bose, P., Demaine, E.D., Lubiw, A.: Efficient algorithms for Petersen\u2019s matching theorem. J. Algorithms 38, 110\u2013134 (2001)","journal-title":"J. Algorithms"},{"issue":"2","key":"10221_CR2","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1017\/S030500410002168X","volume":"37","author":"RL Brooks","year":"1941","unstructured":"Brooks, R.L., Tutte, W.T.: On colouring the nodes of a network. Proc. Camb. Philos. Soc. 37(2), 194 (1941)","journal-title":"Proc. Camb. Philos. Soc."},{"key":"10221_CR3","doi-asserted-by":"publisher","first-page":"449","DOI":"10.4153\/CJM-1965-045-4","volume":"17","author":"J Edmonds","year":"1965","unstructured":"Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449\u2013467 (1965)","journal-title":"Can. J. Math."},{"key":"10221_CR4","doi-asserted-by":"crossref","unstructured":"Micali, S., Vazirani, V.V.: An $$O(\\sqrt{|v|} \\cdot |E|)$$ algorithm for finding maximum matching in general graphs. In: Proceedings of the 21st Annual Symposium on Foundations of Computer Science, pp. 17\u201327, Syracuse, New York, October 1980","DOI":"10.1109\/SFCS.1980.12"},{"key":"10221_CR5","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/BF02392606","volume":"15","author":"J Petersen","year":"1891","unstructured":"Petersen, J.: Die Theorie der regul\u00e4ren graphs. Acta Math. 15, 193\u2013220 (1891)","journal-title":"Acta Math."},{"key":"10221_CR6","doi-asserted-by":"crossref","unstructured":"Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four-colour theorem. J. Comb. Theory Ser. B 70(1), 2\u201344 (1997)","DOI":"10.1006\/jctb.1997.1750"},{"key":"10221_CR7","doi-asserted-by":"crossref","unstructured":"Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pp. 216\u2013226. ACM (1978)","DOI":"10.1145\/800133.804350"}],"container-title":["Theory of Computing Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00224-025-10221-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00224-025-10221-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00224-025-10221-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T10:27:38Z","timestamp":1751970458000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00224-025-10221-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,26]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["10221"],"URL":"https:\/\/doi.org\/10.1007\/s00224-025-10221-2","relation":{},"ISSN":["1432-4350","1433-0490"],"issn-type":[{"type":"print","value":"1432-4350"},{"type":"electronic","value":"1433-0490"}],"subject":[],"published":{"date-parts":[[2025,4,26]]},"assertion":[{"value":"2 April 2025","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 April 2025","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"22"}}