{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T06:59:49Z","timestamp":1775717989401,"version":"3.50.1"},"reference-count":8,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T00:00:00Z","timestamp":1770681600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T00:00:00Z","timestamp":1770681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    An injective edge-coloring of a graph\n                    <jats:italic>G<\/jats:italic>\n                    is an edge-coloring\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\phi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03d5<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of\n                    <jats:italic>G<\/jats:italic>\n                    , not necessarily proper, such that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\phi (e_1) \\ne \\phi (e_2)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03d5<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>e<\/mml:mi>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mi>\u03d5<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>e<\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    whenever there exists an edge\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$e_3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>e<\/mml:mi>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    adjacent to both\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$e_1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>e<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$e_2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>e<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . In particular, edges in a triangle must receive distinct colors. The minimum number of colors needed in an injective edge-coloring of\n                    <jats:italic>G<\/jats:italic>\n                    , denoted by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\chi _{inj}^{\\prime }(G)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mi>\u03c7<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mi>inj<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msubsup>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , is called the injective chromatic index of\n                    <jats:italic>G<\/jats:italic>\n                    . In this paper, we study\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$K_4$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -minor free graphs, or equivalently, graphs with treewidth at most 2. Axenovich et al. [1] studied the injective chromatic index of graphs with treewidth at most\n                    <jats:italic>k<\/jats:italic>\n                    , and their result implies that every\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$K_4$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -minor free graph has injective chromatic index at most 9. This improves a linear bound established by Lv et al. [6] to a constant. For subcubic graphs, Kostochka et al. [5] showed that every subcubic planar graph has injective chromatic index at most 6, which implies that every subcubic\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$K_4$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -minor free graph has injective chromatic index at most 6. In the paper, we study\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$K_4$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -minor free graphs with maximum degree at most 4 and prove that every such graph has injective chromatic index at most 7.\n                  <\/jats:p>","DOI":"10.1007\/s00373-026-03015-x","type":"journal-article","created":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T05:50:17Z","timestamp":1770702617000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Injective Edge-Colorings of $$K_4$$-Minor Free Graphs"],"prefix":"10.1007","volume":"42","author":[{"given":"Fuxiang","family":"Yu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0662-4792","authenticated-orcid":false,"given":"Xiangqian","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,2,10]]},"reference":[{"key":"3015_CR1","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1016\/j.disc.2018.10.018","volume":"342","author":"M Axenovich","year":"2019","unstructured":"Axenovich, M., D\u00f6rr, P., Rollin, J., Urckerdt, T.: Induced and weak induced arboricities. 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