{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:34:45Z","timestamp":1760229285683,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,8]],"date-time":"2022-06-08T00:00:00Z","timestamp":1654646400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSFC","award":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"],"award-info":[{"award-number":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"]}]},{"name":"Science and Technology Commission of Shanghai Municipality","award":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"],"award-info":[{"award-number":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"]}]},{"name":"PhD Scientific Research Foundation of Jiangxi Science and Technology Normal University, China","award":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"],"award-info":[{"award-number":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"]}]},{"name":"NSFC","award":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"],"award-info":[{"award-number":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"]}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"publisher","award":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"],"award-info":[{"award-number":["12071048","12161141006","18dz2271000","2021BSQD34","12161141006","12031018","11771402"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many symmetric properties are well-explored in graph theory, especially in graph coloring, such as symmetric graphs defined by the automorphism groups, symmetric drawing of planar graphs, and symmetric functions which are used to count the number of specific colorings of a graph. This paper is devoted to studying the star edge coloring of 1-planar graphs. The star chromatic index \u03c7st\u2032(G) of a graph G is defined as the smallest k for which the edges of G can be colored by using k colors so that no two adjacent edges get the same color and no bichromatic paths or cycles of length four are produced. A graph G is called 1-planar if it can be drawn in the plane such that each edge crosses at most one other edge. In this paper, we prove that every 1-planar graph G satisfies \u03c7st\u2032(G)\u22647.75\u0394+166; and moreover \u03c7st\u2032(G)\u2264\u230a1.5\u0394\u230b+500 if G contains no 4-cycles, and \u03c7st\u2032(G)\u22642.75\u0394+116 if G is 3-connected, or optimal, or NIC-planar.<\/jats:p>","DOI":"10.3390\/sym14061177","type":"journal-article","created":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T02:01:44Z","timestamp":1655085704000},"page":"1177","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Star Chromatic Index of 1-Planar Graphs"],"prefix":"10.3390","volume":"14","author":[{"given":"Yiqiao","family":"Wang","sequence":"first","affiliation":[{"name":"School of Management, Beijing University of Chinese Medicine, Beijing 100029, China"}]},{"given":"Juan","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang 330038, China"}]},{"given":"Yongtang","family":"Shi","sequence":"additional","affiliation":[{"name":"Center for Combinatorics and LPMC, Nankai University, Tianjin 300071, China"}]},{"given":"Weifan","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1006\/aima.1995.1020","article-title":"A symmetric function generalization of the chromatic polynomial of a graph","volume":"111","author":"Stanley","year":"1995","journal-title":"Adv. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"105572","DOI":"10.1016\/j.jcta.2021.105572","article-title":"Modular relations of the Tutte symmetric function","volume":"187","author":"Crew","year":"2022","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Gross, J.L., Yellen, J., and Anderson, M. (2018). Graph Theory and Its Applications, Chapman and Hall\/CRC. [3rd ed.].","DOI":"10.1201\/9780429425134"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1006\/jctb.1997.1724","article-title":"A bound on the strong chromatic index of a graph","volume":"69","author":"Molloy","year":"1997","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.jctb.2022.01.009","article-title":"Colouring graphs with sparse neighbourhoods: Bounds and applications","volume":"155","author":"Bonamy","year":"2022","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_6","first-page":"205","article-title":"The strong chromatic index of graphs","volume":"29","author":"Faudree","year":"1990","journal-title":"Ars Comb."},{"key":"ref_7","first-page":"98","article-title":"An upper bound on the star chromatic index of graphs with \u0394\u22657","volume":"44","author":"Liu","year":"2008","journal-title":"J. Lanzhou Univ. Nat. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1002\/jgt.21644","article-title":"Star chromatic index","volume":"72","author":"Mohar","year":"2013","journal-title":"J. Graph Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1002\/jgt.22402","article-title":"Note on list star edge-coloring of subcubic graphs","volume":"90","year":"2019","journal-title":"J. Graph Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1002\/jgt.22230","article-title":"Star chromatic index of subcubic multigraphs","volume":"88","author":"Lei","year":"2018","journal-title":"J. Graph Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1002\/jgt.21862","article-title":"Star edge coloring of some classes of graphs","volume":"81","year":"2016","journal-title":"J. Graph Theory"},{"key":"ref_12","first-page":"480","article-title":"Edge-partition and star chromatic index","volume":"333","author":"Wang","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1016\/j.disc.2005.11.056","article-title":"The structure of 1-planar graphs","volume":"307","author":"Fabrici","year":"2007","journal-title":"Discret. Math."},{"key":"ref_14","unstructured":"Wang, Y., Wang, W., Song, N., and Wang, J. (2022). The strong chromatic index of 1-planar graphs. arXiv Preprint, Available online: https:\/\/arxiv.org\/pdf\/2205.14680.pdf."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.dam.2014.05.025","article-title":"A note on 1-planar graphs","volume":"175","author":"Ackerman","year":"2014","journal-title":"Discret. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.tcs.2017.12.024","article-title":"New results on edge partitions of 1-plane graphs","volume":"713","author":"Didimo","year":"2018","journal-title":"Theor. Comput. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.tcs.2016.12.004","article-title":"On partitioning the edges of 1-plane graphs","volume":"662","author":"Lenhart","year":"2017","journal-title":"Theor. Comput. Sci."},{"key":"ref_18","first-page":"2345","article-title":"Ortho-polygon visibility representations of embedded graphs","volume":"80","author":"Didimo","year":"2017","journal-title":"Algorithmica"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.dam.2019.05.001","article-title":"Light structures in 1-planar graphs with an application to linear 2-arboricity","volume":"267","author":"Liu","year":"2019","journal-title":"Discret. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Liu, J., Wang, Y., and Wang, W. (2022). Light edges in 1-planar graphs. J. Graph Theory, accepted for publication.","DOI":"10.1002\/jgt.22851"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1907","DOI":"10.1007\/s40840-016-0434-z","article-title":"Linear 2-arboricity of toroidal graphs","volume":"41","author":"Wang","year":"2018","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_22","first-page":"25","article-title":"On an estimate of the chromatic index of a p-graph","volume":"3","author":"Vizing","year":"1964","journal-title":"Diskret. Analiz."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.ipl.2010.11.001","article-title":"On edge colorings of 1-planar graphs","volume":"111","author":"Zhang","year":"2011","journal-title":"Inform. Process. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"87","DOI":"10.7151\/dmgt.1434","article-title":"Decompositions of quadrangle-free planar graphs","volume":"29","author":"Borodin","year":"2009","journal-title":"Discuss. Math. Graph Theory"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"112351","DOI":"10.1016\/j.disc.2021.112351","article-title":"Equitable partition of planar graphs","volume":"344","author":"Kim","year":"2021","journal-title":"Discret. Math."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Tomescu, M.A., Ja\u00a8ntschi, L., and Rotaru, D.I. (2021). Figures of graph partitioning by counting, sequence and layer matrices. Mathematics, 9.","DOI":"10.3390\/math9121419"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1177\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:25:49Z","timestamp":1760138749000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1177"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,8]]},"references-count":26,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061177"],"URL":"https:\/\/doi.org\/10.3390\/sym14061177","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,6,8]]}}}