{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T16:07:04Z","timestamp":1776874024645,"version":"3.51.2"},"reference-count":23,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2024,5,12]],"date-time":"2024-05-12T00:00:00Z","timestamp":1715472000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/http\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009935","name":"Indo-French Centre for Applied Mathematics","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100009935","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We say that a signed graph is \u2010<jats:italic>critical<\/jats:italic> if it is not \u2010colorable but every one of its proper subgraphs is \u2010colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every 3\u2010critical signed graph on  vertices has at least  edges, and that this bound is asymptotically tight. It follows that every signed planar or projective\u2010planar graph of girth at least 6 is (circular) 3\u2010colorable, and for the projective\u2010planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph , which is the positive triangle augmented with a negative loop on each vertex.<\/jats:p>","DOI":"10.1002\/jgt.23117","type":"journal-article","created":{"date-parts":[[2024,5,13]],"date-time":"2024-05-13T05:13:16Z","timestamp":1715577196000},"page":"212-239","update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Density of 3\u2010critical signed graphs"],"prefix":"10.1002","volume":"107","author":[{"given":"Laurent","family":"Beaudou","sequence":"first","affiliation":[{"name":"Universit\u00e9 Clermont Auvergne Clermont\u2010Ferrand France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Penny","family":"Haxell","sequence":"additional","affiliation":[{"name":"University of Waterloo Waterloo Ontario Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0009-0003-7155-4967","authenticated-orcid":false,"given":"Kathryn","family":"Nurse","sequence":"additional","affiliation":[{"name":"Simon Fraser University Burnaby Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sagnik","family":"Sen","sequence":"additional","affiliation":[{"name":"Indian Institute of Technology Dharwad India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0003-4722-4880","authenticated-orcid":false,"given":"Zhouningxin","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and LPMC Nankai University Tianjin China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2024,5,12]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/19M1242513"},{"key":"e_1_2_7_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-016-3356-3"},{"key":"e_1_2_7_4_1","first-page":"109","article-title":"Ein Dreifarbensatz f\u00fc r dreikreisfreie Netze auf der Kugel","volume":"8","author":"Gr\u00f6tzsch H.","year":"1959","journal-title":"Wiss. Z. Martin\u2010Luther\u2010Univ. Halle\u2010Wittenberg Math.\u2010Natur. Reihe"},{"key":"e_1_2_7_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2018.01.002"},{"key":"e_1_2_7_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-86893-0.50031-0"},{"key":"e_1_2_7_7_1","first-page":"71","volume-title":"Selected topics in graph theory","author":"Jaeger F.","year":"1988"},{"key":"e_1_2_7_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2020.103215"},{"key":"e_1_2_7_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/21M1436257"},{"key":"e_1_2_7_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2014.05.002"},{"key":"e_1_2_7_11_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-014-3020-x"},{"key":"e_1_2_7_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2013.06.003"},{"key":"e_1_2_7_13_1","doi-asserted-by":"publisher","DOI":"10.37236\/4938"},{"key":"e_1_2_7_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2021.11.002"},{"key":"e_1_2_7_15_1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.21817"},{"key":"e_1_2_7_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2020.103222"},{"key":"e_1_2_7_17_1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22992"},{"key":"e_1_2_7_18_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2023.113604"},{"key":"e_1_2_7_19_1","doi-asserted-by":"publisher","DOI":"10.37236\/9938"},{"key":"e_1_2_7_20_1","volume-title":"The four color problem","author":"Ore O.","year":"1967"},{"key":"e_1_2_7_21_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-020-4177-y"},{"key":"e_1_2_7_22_1","volume-title":"Introduction to graph theory","author":"West D.","year":"1996"},{"key":"e_1_2_7_23_1","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(82)90033-6"},{"key":"e_1_2_7_24_1","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(82)90144-3"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.23117","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T15:42:48Z","timestamp":1720712568000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.23117"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,12]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["10.1002\/jgt.23117"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1002\/jgt.23117","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,12]]},"assertion":[{"value":"2023-04-16","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-04-22","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-05-12","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}