{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T03:21:41Z","timestamp":1765164101626,"version":"3.46.0"},"reference-count":8,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T00:00:00Z","timestamp":1760486400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T00:00:00Z","timestamp":1760486400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100009087","name":"Industrial University of Santander","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100009087","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    A coloring on a finite or countable set\n                    <jats:italic>X<\/jats:italic>\n                    is a function\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varphi : [X]^{2} \\rightarrow \\{0,1\\}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mo>[<\/mml:mo>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:mo>]<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\u2192<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                              <mml:mn>0<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>}<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$[X]^{2}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is the collection of unordered pairs of\n                    <jats:italic>X<\/jats:italic>\n                    . The collection of homogeneous sets for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varphi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c6<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , denoted by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\operatorname {hom}(\\varphi )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>hom<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , consists of all\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$H \\subseteq X$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo>\u2286<\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varphi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c6<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is constant on\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$[H]^2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ; clearly,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\operatorname {hom}(\\varphi ) = \\operatorname {hom}(1-\\varphi )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>hom<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>hom<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . A coloring\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varphi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c6<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is\n                    <jats:italic>reconstructible<\/jats:italic>\n                    up to complementation from its homogeneous sets if, for any coloring\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\psi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c8<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    on\n                    <jats:italic>X<\/jats:italic>\n                    such that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\operatorname {hom}(\\varphi ) = \\operatorname {hom}(\\psi )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>hom<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>hom<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , either\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\psi = \\varphi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    or\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\psi = 1-\\varphi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . By\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {R}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>R<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    we denote the collection of all colorings reconstructible from their homogeneous sets. Let\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varphi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c6<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\psi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c8<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    be colorings on\n                    <jats:italic>X<\/jats:italic>\n                    , and set\n                    <jats:disp-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$ D(\\varphi , \\psi ) = \\{ \\{x,y\\} \\in [X]^2: \\; \\psi \\{x,y\\} \\ne \\varphi \\{x,y\\}\\}. $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>\u03c6<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>\u03c8<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>{<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mo>}<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mo>[<\/mml:mo>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:mo>]<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mspace\/>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mo>}<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mo>}<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>}<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:disp-formula>\n                    If\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varphi \\not \\in \\mathcal {R}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>\u2209<\/mml:mo>\n                            <mml:mi>R<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , let\n                    <jats:disp-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$ r(\\varphi ) = \\min \\{|D(\\varphi , \\psi )|: \\; \\operatorname {hom}(\\varphi ) = \\operatorname {hom}(\\psi ), \\, \\psi \\ne \\varphi , \\, \\psi \\ne 1-\\varphi \\}. $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>min<\/mml:mo>\n                            <mml:mo>{<\/mml:mo>\n                            <mml:mo>|<\/mml:mo>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>|<\/mml:mo>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mspace\/>\n                            <mml:mo>hom<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>hom<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace\/>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace\/>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>}<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:disp-formula>\n                    A coloring\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\psi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c8<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\operatorname {hom}(\\varphi )=\\operatorname {hom}(\\psi )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>hom<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>hom<\/mml:mo>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varphi \\ne \\psi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mi>\u03c8<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$1-\\varphi \\ne \\psi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mi>\u03c8<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is called a\n                    <jats:italic>non trivial reconstruction<\/jats:italic>\n                    of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varphi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c6<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . If, in addition,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$r(\\varphi ) =|D(\\varphi , \\psi )|$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>|<\/mml:mo>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\u03c8<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                            <mml:mo>|<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , we call\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\psi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c8<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    a\n                    <jats:italic>minimal reconstruction<\/jats:italic>\n                    of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varphi $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c6<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . The purpose of this article is to study the minimal reconstructions of a coloring. The main result is that, for sufficiently large\n                    <jats:italic>X<\/jats:italic>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$r(\\varphi )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c6<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    can only take the values 1 or 4.\n                  <\/jats:p>","DOI":"10.1007\/s00373-025-02967-w","type":"journal-article","created":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T17:33:28Z","timestamp":1760549608000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Minimal Reconstructions of a Coloring from its Homogeneous Sets"],"prefix":"10.1007","volume":"41","author":[{"given":"Diego","family":"Gamboa","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3452-2312","authenticated-orcid":false,"given":"Carlos","family":"Uzc\u00e1tegui-Aylwin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,10,15]]},"reference":[{"key":"2967_CR1","doi-asserted-by":"crossref","unstructured":"Bondy, J.A.: A graph reconstructor\u2019s manual. 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