{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:15:31Z","timestamp":1757618131953,"version":"3.44.0"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,6,2]],"date-time":"2025-06-02T00:00:00Z","timestamp":1748822400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,6,2]],"date-time":"2025-06-02T00:00:00Z","timestamp":1748822400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"South African National Research Foundation","award":["132588, 129265"],"award-info":[{"award-number":["132588, 129265"]}]},{"DOI":"10.13039\/501100006565","name":"University of Johannesburg","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006565","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":[[2025,8]]},"abstract":"Abstract<\/jats:title>\n A set S<\/jats:italic> of vertices in a graph G<\/jats:italic> is a dominating set of G<\/jats:italic> if every vertex not in S<\/jats:italic> has a neighbor in S<\/jats:italic>, where two vertices are neighbors if they are adjacent. If G<\/jats:italic> is isolate-free, then a set \n \n $$S \\subseteq V(G)$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n \u2286<\/mml:mo>\n V<\/mml:mi>\n (<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a double dominating set of G<\/jats:italic> if every vertex in \n \n $$V(G) \\setminus S$$<\/jats:tex-math>\n \n \n V<\/mml:mi>\n (<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n \\<\/mml:mo>\n S<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> has at least two neighbors in S<\/jats:italic>, and every vertex in S<\/jats:italic> has at least one neighbor in S<\/jats:italic>. A double coalition in G<\/jats:italic> consists of two disjoint sets of vertices X<\/jats:italic> and Y<\/jats:italic> of G<\/jats:italic>, neither of which is a double dominating set but whose union \n \n $$X \\cup Y$$<\/jats:tex-math>\n \n \n X<\/mml:mi>\n \u222a<\/mml:mo>\n Y<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a double dominating set of G<\/jats:italic>. Such sets X<\/jats:italic> and Y<\/jats:italic> are said to form a double coalition. A double coalition partition in G<\/jats:italic> is a vertex partition \n \n $$\\Psi = \\{V_1,V_2,\\ldots ,V_k\\}$$<\/jats:tex-math>\n \n \n \u03a8<\/mml:mi>\n =<\/mml:mo>\n {<\/mml:mo>\n \n V<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n V<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n V<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> such that for all \n \n $$i \\in [k]$$<\/jats:tex-math>\n \n \n i<\/mml:mi>\n \u2208<\/mml:mo>\n [<\/mml:mo>\n k<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the set \n \n $$V_i$$<\/jats:tex-math>\n \n \n V<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> forms a double coalition with another set \n \n $$V_j$$<\/jats:tex-math>\n \n \n V<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for some j<\/jats:italic>, where \n \n $$j \\in [k] \\setminus \\{i\\}$$<\/jats:tex-math>\n \n \n j<\/mml:mi>\n \u2208<\/mml:mo>\n [<\/mml:mo>\n k<\/mml:mi>\n ]<\/mml:mo>\n \\<\/mml:mo>\n {<\/mml:mo>\n i<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. The double coalition number, \n \n $$\\textrm{DC}(G)$$<\/jats:tex-math>\n \n \n DC<\/mml:mtext>\n (<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, of G<\/jats:italic> equals the maximum order of a double coalition partition in G<\/jats:italic>. We discuss the problem to determine or estimate the best possible constants \n \n $$\\theta _{r}^{\\textrm{reg}}$$<\/jats:tex-math>\n \n \n \u03b8<\/mml:mi>\n \n r<\/mml:mi>\n <\/mml:mrow>\n reg<\/mml:mtext>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$\\theta _{r}$$<\/jats:tex-math>\n \n \n \u03b8<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> (which depend only on\u00a0r<\/jats:italic>) for each \n \n $$r \\ge 3$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n \u2265<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, such that \n \n $$\\textrm{DC}(G) \\le \\theta _{r}^{\\textrm{reg}} \\times r$$<\/jats:tex-math>\n \n \n DC<\/mml:mtext>\n \n (<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2264<\/mml:mo>\n \n \u03b8<\/mml:mi>\n \n r<\/mml:mi>\n <\/mml:mrow>\n reg<\/mml:mtext>\n <\/mml:msubsup>\n \u00d7<\/mml:mo>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for the class of r<\/jats:italic>-regular graphs G<\/jats:italic> and \n \n $$\\textrm{DC}(G) \\le \\theta _{r} \\times \\Delta (G)$$<\/jats:tex-math>\n \n \n DC<\/mml:mtext>\n \n (<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2264<\/mml:mo>\n \n \u03b8<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \u00d7<\/mml:mo>\n \u0394<\/mml:mi>\n \n (<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for the class of graphs G<\/jats:italic> with minimum degree equal to\u00a0r<\/jats:italic>. We show that \n \n $$\\theta _{r}^{\\textrm{reg}} \\ge 2 \\left( \\frac{r-1}{r} \\right) $$<\/jats:tex-math>\n \n \n \n \u03b8<\/mml:mi>\n \n r<\/mml:mi>\n <\/mml:mrow>\n reg<\/mml:mtext>\n <\/mml:msubsup>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n \n \n \n r<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n r<\/mml:mi>\n <\/mml:mfrac>\n <\/mml:mfenced>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for all \n \n $$r \\ge 3$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n \u2265<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and that equality holds if \n \n $$r \\in \\{3,4\\}$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n \u2208<\/mml:mo>\n {<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n 4<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, while \n \n $$\\theta _{r} \\ge 2$$<\/jats:tex-math>\n \n \n \n \u03b8<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for all \n \n $$r \\ge 3$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n \u2265<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and \n \n $$\\theta _{r} \\ge 3$$<\/jats:tex-math>\n \n \n \n \u03b8<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \u2265<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for r<\/jats:italic> sufficiently large. Moreover, we show that \n \n $$\\theta _3 = 2$$<\/jats:tex-math>\n \n \n \n \u03b8<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Finally, we prove that \n \n $$5 \\le \\textrm{DC}(G) \\le 6$$<\/jats:tex-math>\n \n \n 5<\/mml:mn>\n \u2264<\/mml:mo>\n DC<\/mml:mtext>\n (<\/mml:mo>\n G<\/mml:mi>\n )<\/mml:mo>\n \u2264<\/mml:mo>\n 6<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> whenever G<\/jats:italic> is a 4-regular graph.<\/jats:p>","DOI":"10.1007\/s00373-025-02937-2","type":"journal-article","created":{"date-parts":[[2025,6,2]],"date-time":"2025-06-02T14:45:30Z","timestamp":1748875530000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Double Coalitions in Regular Graphs"],"prefix":"10.1007","volume":"41","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8185-067X","authenticated-orcid":false,"given":"Michael A.","family":"Henning","sequence":"first","affiliation":[]},{"given":"Doost Ali","family":"Mojdeh","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,6,2]]},"reference":[{"issue":"11","key":"2937_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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