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A total coalition in\n                    <jats:italic>G<\/jats:italic>\n                    consists of two disjoint sets of vertices\n                    <jats:italic>X<\/jats:italic>\n                    and\n                    <jats:italic>Y<\/jats:italic>\n                    of\n                    <jats:italic>G<\/jats:italic>\n                    , neither of which is a total dominating set but whose union\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$X \\cup Y$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo>\u222a<\/mml:mo>\n                            <mml:mi>Y<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a total dominating set of\n                    <jats:italic>G<\/jats:italic>\n                    . Such sets\n                    <jats:italic>X<\/jats:italic>\n                    and\n                    <jats:italic>Y<\/jats:italic>\n                    are said to form a total coalition. A total coalition partition in\n                    <jats:italic>G<\/jats:italic>\n                    is a vertex partition\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\Psi = \\{V_1,V_2,\\ldots ,V_k\\}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03a8<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>{<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>V<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>V<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\u2026<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>V<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>}<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that for all\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$i \\in [k]$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:mo>[<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>]<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , the set\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$V_i$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>V<\/mml:mi>\n                            <mml:mi>i<\/mml:mi>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    forms a total coalition with another set\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$V_j$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>V<\/mml:mi>\n                            <mml:mi>j<\/mml:mi>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for some\n                    <jats:italic>j<\/jats:italic>\n                    , where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$j \\in [k] \\setminus \\{i\\}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:mo>[<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>]<\/mml:mo>\n                            <mml:mo>\\<\/mml:mo>\n                            <mml:mo>{<\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>}<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We emphasize that none of the sets in\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>\u03a8<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a total dominating set of\n                    <jats:italic>G<\/jats:italic>\n                    . The total coalition number\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$C_t(G)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>C<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\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                    in\n                    <jats:italic>G<\/jats:italic>\n                    equals the maximum order of a total coalition partition in\n                    <jats:italic>G<\/jats:italic>\n                    . We study total coalitions in claw-free cubic graphs with certain structural properties, namely, graphs containing double-bonded triangle-units, that is, two vertex disjoint triangles joined by two edges.\n                  <\/jats:p>","DOI":"10.1007\/s00373-026-03012-0","type":"journal-article","created":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T05:19:46Z","timestamp":1768972786000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Total Coalitions in Claw-Free Cubic Graphs Containing Double-Bonded Triangle-Units"],"prefix":"10.1007","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1877-9983","authenticated-orcid":false,"given":"Zolt\u00e1n L.","family":"Bl\u00e1zsik","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8185-067X","authenticated-orcid":false,"given":"Michael A.","family":"Henning","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8603-8533","authenticated-orcid":false,"given":"Shahin N.","family":"Jogan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,1,21]]},"reference":[{"issue":"11","key":"3012_CR1","doi-asserted-by":"publisher","first-page":"2283","DOI":"10.2989\/16073606.2024.2365365","volume":"47","author":"S Alikhani","year":"2024","unstructured":"Alikhani, S., Bakhshesh, D., Golmohammadi, H.: Total coalitions in graphs. 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