{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T19:26:13Z","timestamp":1776540373287,"version":"3.51.2"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T00:00:00Z","timestamp":1773446400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T00:00:00Z","timestamp":1773446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["FKZ AX 93\/2-1"],"award-info":[{"award-number":["FKZ AX 93\/2-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2026,4]]},"abstract":"Abstract<\/jats:title>\n \n A subset\n M<\/jats:italic>\n of vertices in a graph\n G<\/jats:italic>\n is a mutual-visibility set if any two vertices\n u<\/jats:italic>\n and\n v<\/jats:italic>\n in\n M<\/jats:italic>\n \u201csee\u201d each other in\n G<\/jats:italic>\n , that is, there exists a shortest\n u<\/jats:italic>\n ,\u00a0\n v<\/jats:italic>\n -path in\n G<\/jats:italic>\n that contains no elements of\n M<\/jats:italic>\n as internal vertices. The mutual-visibility number\n \n \n $$\\mu (G)$$<\/jats:tex-math>\n \n \n \u03bc<\/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>\n of a graph\n G<\/jats:italic>\n is the largest size of a mutual-visibility set in\n G<\/jats:italic>\n . Let\n \n \n $$n\\in \\mathbb {N}$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$Q_{n}$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be an\n n<\/jats:italic>\n -dimensional hypercube. Cicerone, Di Fonso, Di Stefano, Navarra, and Piselli showed that\n \n \n $$2^{n}\/\\sqrt{n}\\le \\mu (Q_{n})\\le 2^{n-1}$$<\/jats:tex-math>\n \n \n \n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n \/<\/mml:mo>\n \n n<\/mml:mi>\n <\/mml:msqrt>\n \u2264<\/mml:mo>\n \u03bc<\/mml:mi>\n \n (<\/mml:mo>\n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2264<\/mml:mo>\n \n 2<\/mml:mn>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In this paper, we prove that\n \n \n $$\\mu (Q_{n})>0.186\\cdot 2^n$$<\/jats:tex-math>\n \n \n \u03bc<\/mml:mi>\n \n (<\/mml:mo>\n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n 0.186<\/mml:mn>\n \u00b7<\/mml:mo>\n \n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and thus establish that\n \n \n $$\\mu (Q_{n})=\\Theta (2^{n})$$<\/jats:tex-math>\n \n \n \u03bc<\/mml:mi>\n \n (<\/mml:mo>\n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \u0398<\/mml:mi>\n \n (<\/mml:mo>\n \n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We also consider the chromatic mutual-visibility number,\n \n \n $$\\chi _{\\mu }(G)$$<\/jats:tex-math>\n \n \n \n \u03c7<\/mml:mi>\n \u03bc<\/mml:mi>\n <\/mml:msub>\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>\n , defined as the smallest number of colors used on vertices of\n G<\/jats:italic>\n , such that every color class is a mutual-visibility set in\n G<\/jats:italic>\n . Klav\u017ear, Kuziak, Valenzuela-Tripodoro, and Yero asked whether\n \n \n $$\\chi _{\\mu }(Q_{n})=O(1)$$<\/jats:tex-math>\n \n \n \n \u03c7<\/mml:mi>\n \u03bc<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n O<\/mml:mi>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We answer their question in the negative, namely, we show that\n \n \n $$\\chi _{\\mu }(Q_{n})$$<\/jats:tex-math>\n \n \n \n \u03c7<\/mml:mi>\n \u03bc<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a growing function of\n n<\/jats:italic>\n . Moreover, we show that\n \n \n $$\\chi _{\\mu }(Q_{n})=O(\\log \\log {n})$$<\/jats:tex-math>\n \n \n \n \u03c7<\/mml:mi>\n \u03bc<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n O<\/mml:mi>\n \n (<\/mml:mo>\n log<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Finally, we study the so-called total mutual-visibility number of graphs and give asymptotically tight bounds on this parameter for hypercubes.\n <\/jats:p>","DOI":"10.1007\/s00373-026-03025-9","type":"journal-article","created":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T12:47:28Z","timestamp":1773492448000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Visibility in Hypercubes"],"prefix":"10.1007","volume":"42","author":[{"given":"Maria","family":"Axenovich","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0008-5439-5390","authenticated-orcid":false,"given":"Dingyuan","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,14]]},"reference":[{"key":"3025_CR1","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1007\/s11083-016-9399-7","volume":"34","author":"M Axenovich","year":"2017","unstructured":"Axenovich, M., Walzer, S.: Boolean lattices: Ramsey properties and embeddings. 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