{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,5]],"date-time":"2025-06-05T21:27:36Z","timestamp":1749158856566,"version":"3.37.3"},"reference-count":7,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2024,8,18]],"date-time":"2024-08-18T00:00:00Z","timestamp":1723939200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,8,18]],"date-time":"2024-08-18T00:00:00Z","timestamp":1723939200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Dr.","award":["132696"],"award-info":[{"award-number":["132696"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2024,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {H}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>H<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be a set of graphs. The planar Tur\u00e1n number, <jats:inline-formula><jats:alternatives><jats:tex-math>$$ex_\\mathcal {P}(n,\\mathcal {H})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>e<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, is the maximum number of edges in an <jats:italic>n<\/jats:italic>-vertex planar graph which does not contain any member of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {H}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>H<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as a subgraph. When <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {H}=\\{H\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has only one element, we usually write <jats:inline-formula><jats:alternatives><jats:tex-math>$$ex_\\mathcal {P}(n,H)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>e<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> instead. The study of extremal planar graphs was initiated by Dowden (J Graph Theory 83(3):213\u2013230, 2016). He obtained sharp upper bounds for both <jats:inline-formula><jats:alternatives><jats:tex-math>$$ex_\\mathcal {P}(n,C_5)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>e<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>C<\/mml:mi>\n                        <mml:mn>5<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$ex_\\mathcal {P}(n,K_4)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>e<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>K<\/mml:mi>\n                        <mml:mn>4<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Later on, sharp upper bounds were proved for <jats:inline-formula><jats:alternatives><jats:tex-math>$$ex_\\mathcal {P}(n,C_6)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>e<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>C<\/mml:mi>\n                        <mml:mn>6<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$ex_\\mathcal {P}(n,C_7)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>e<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>C<\/mml:mi>\n                        <mml:mn>7<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this paper, we show that <jats:inline-formula><jats:alternatives><jats:tex-math>$$ex_\\mathcal {P}(n,\\{K_4,C_5\\})\\le {15\\over 7}(n-2)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>e<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mo>{<\/mml:mo>\n                        <mml:msub>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:mn>4<\/mml:mn>\n                        <\/mml:msub>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:msub>\n                          <mml:mi>C<\/mml:mi>\n                          <mml:mn>5<\/mml:mn>\n                        <\/mml:msub>\n                        <mml:mo>}<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mfrac>\n                      <mml:mn>15<\/mml:mn>\n                      <mml:mn>7<\/mml:mn>\n                    <\/mml:mfrac>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$ex_\\mathcal {P}(n,\\{K_4,C_6\\})\\le {7\\over 3}(n-2)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>e<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>P<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mo>{<\/mml:mo>\n                        <mml:msub>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:mn>4<\/mml:mn>\n                        <\/mml:msub>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:msub>\n                          <mml:mi>C<\/mml:mi>\n                          <mml:mn>6<\/mml:mn>\n                        <\/mml:msub>\n                        <mml:mo>}<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mfrac>\n                      <mml:mn>7<\/mml:mn>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:mfrac>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We also give constructions which show the bounds are sharp for infinitely many <jats:italic>n<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s00373-024-02830-4","type":"journal-article","created":{"date-parts":[[2024,8,18]],"date-time":"2024-08-18T10:01:48Z","timestamp":1723975308000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Planar Tur\u00e1n Number of $$\\{K_4,C_5\\}$$ and $$\\{K_4,C_6\\}$$"],"prefix":"10.1007","volume":"40","author":[{"given":"Ervin","family":"Gy\u0151ri","sequence":"first","affiliation":[]},{"given":"Alan","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0005-7525-3541","authenticated-orcid":false,"given":"Runtian","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,18]]},"reference":[{"issue":"3","key":"2830_CR1","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1002\/jgt.21991","volume":"83","author":"C Dowden","year":"2016","unstructured":"Dowden, C.: Extremal $$C_4$$-free\/$$C_5$$-free planar graphs. 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Math."},{"key":"2830_CR5","unstructured":"Gy\u0151ri, E., Li, A., Zhou, R.: The planar Tur\u00e1n number of the seven-cycle (2023). arXiv:2307.06909"},{"issue":"12","key":"2830_CR6","volume":"342","author":"Y Lan","year":"2019","unstructured":"Lan, Y., Shi, Y., Song, Z.X.: Extremal theta-free planar graphs. Disc. Math. 342(12), 111610 (2019)","journal-title":"Extremal theta-free planar graphs. Disc. Math."},{"key":"2830_CR7","unstructured":"Shi, R., Walsh, Z., Yu, X.: Planar Tur\u00e1n number of the 7-cycle (2023). arXiv:2306.13594"}],"container-title":["Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-024-02830-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00373-024-02830-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-024-02830-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T14:02:29Z","timestamp":1735308149000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00373-024-02830-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,18]]},"references-count":7,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["2830"],"URL":"https:\/\/doi.org\/10.1007\/s00373-024-02830-4","relation":{},"ISSN":["0911-0119","1435-5914"],"issn-type":[{"type":"print","value":"0911-0119"},{"type":"electronic","value":"1435-5914"}],"subject":[],"published":{"date-parts":[[2024,8,18]]},"assertion":[{"value":"23 September 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 August 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 August 2024","order":4,"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 that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}],"article-number":"99"}}