{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T14:59:04Z","timestamp":1783522744156,"version":"3.55.0"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T00:00:00Z","timestamp":1768953600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research Foundation of Xihua University","award":["Z242124"],"award-info":[{"award-number":["Z242124"]}]},{"name":"Scientific Research Foundation of Chengdu University of Information Technology","award":["KYTZ2022146"],"award-info":[{"award-number":["KYTZ2022146"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Dragonfly networks D(n,h) are a class of interconnection topologies widely used for large-scale high-performance computing (HPC) systems. In such networks, path connectivity serves as a fundamental metric for evaluating fault tolerance and operational reliability. Let G be a connected simple graph with vertex set V(G). Let \u03a9 be a subset of V(G) with cardinality at least two. A path containing all vertices of \u03a9 is said to be an \u03a9-path of G. Two paths (T1 and T2) of G are internally disjoint if V(T1)\u2229V(T2)=\u03a9 and E(T1)\u2229E(T2)=\u2205. For an integer with 2\u2264\u2113, the \u2113-path connectivity \u03c0\u2113(G) is defined as \u03c0\u2113(G)=min{\u03c0G(\u03a9)|\u03a9\u2286V(G)and|\u03a9|=\u2113}, where \u03c0G(\u03a9) represents the maximum number of internally disjoint \u03a9-paths. This paper focuses on resolving the exact value of 3-path connectivity of dragonfly networks, \u03c03(D(n,h)), defined as the maximum number of internally disjoint paths among any three distinct vertices in D(n,h). For D(n,h) with n\u22655 and h\u22652, the exact 3-path connectivity is \u03c03(D(n,h))=\u230a3h+2n4\u230b if h\u2264n\u22122, and \u03c03(D(n,h))=\u230a3n+2h\u221224\u230b if h\u2265n\u22121.<\/jats:p>","DOI":"10.3390\/axioms15010076","type":"journal-article","created":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T13:59:54Z","timestamp":1769003994000},"page":"76","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The 3-Path Connectivity of Dragonfly Networks"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3753-3671","authenticated-orcid":false,"given":"Guanlin","family":"He","sequence":"first","affiliation":[{"name":"School of Computer and Software Engineering, Xihua University, Chengdu 610039, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3575-7156","authenticated-orcid":false,"given":"Zengxian","family":"Tian","sequence":"additional","affiliation":[{"name":"College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Whitney, H. (1992). Congruent graphs and the connectivity of graphs. Hassler Whitney Collected Papers, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-2972-8_4"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/0012-365X(86)90068-3","article-title":"Path-connectivity in graphs","volume":"59","author":"Hager","year":"1986","journal-title":"Discret. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1145\/1394608.1382129","article-title":"Technology-driven, highly-scalable dragonfly topology","volume":"36","author":"Kim","year":"2008","journal-title":"ACM SIGARCH Comput. Archit. News"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Yang, X., Jenkins, J., Mubarak, M., Ross, R.B., and Lan, Z. (2016, January 13\u201318). Watch out for the bully! job interference study on dragonfly network. Proceedings of the SC\u201916: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, Salt Lake City, UT, USA.","DOI":"10.1109\/SC.2016.63"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2520","DOI":"10.1109\/TPDS.2015.2503746","article-title":"Deadlock-free broadcast routing in dragonfly networks without virtual channels","volume":"27","author":"Xiang","year":"2015","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1109\/TDSC.2017.2693372","article-title":"Fault-tolerant adaptive routing in dragonfly networks","volume":"16","author":"Xiang","year":"2017","journal-title":"IEEE Trans. Dependable Secur. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Hastings, E., Rincon-Cruz, D., Spehlmann, M., Meyers, S., Xu, A., Bunde, D.P., and Leung, V.J. (2015, January 8\u201311). Comparing global link arrangements for dragonfly networks. Proceedings of the 2015 IEEE International Conference on Cluster Computing, Chicago, IL, USA.","DOI":"10.1109\/CLUSTER.2015.57"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Curtsinger, R., and Bunde, D. (2019, January 16\u201320). Shortest paths in dragonfly systems. Proceedings of the 2019 International Workshop of High-Perfomance Interconnection Networks in the Exascale and Big-Data Era (HiPNEB), Washington, DC, USA.","DOI":"10.1109\/HiPINEB.2019.00008"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1201","DOI":"10.1093\/comjnl\/bxad052","article-title":"Pancyclic and Hamiltonian properties of dragonfly networks","volume":"67","author":"Huo","year":"2024","journal-title":"Comput. J."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Tian, Z., and He, G. (2025). Two-Disjoint-Cycle-Cover Pancyclicity of Dragonfly Networks. Mathematics, 13.","DOI":"10.20944\/preprints202510.1485.v1"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.amc.2016.10.002","article-title":"The generalized 3-connectivity of graph products","volume":"295","author":"Li","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s10878-016-0075-z","article-title":"On minimally 2-connected graphs with generalized connectivity \u03ba3 = 2","volume":"34","author":"Li","year":"2017","journal-title":"J. Comb. Optim."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"882","DOI":"10.1016\/j.amc.2018.11.006","article-title":"The generalized 3-connectivity of the Mycielskian of a graph","volume":"347","author":"Li","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1016\/j.amc.2018.11.038","article-title":"The generalized connectivity of the line graph and the total graph for the complete bipartite graph","volume":"347","author":"Li","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1080\/00207160.2014.987762","article-title":"Path-connectivity of lexicographic product graphs","volume":"93","author":"Mao","year":"2016","journal-title":"Int. J. Comput. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2521","DOI":"10.1007\/s00373-021-02376-9","article-title":"On tree-connectivity and path-connectivity of graphs","volume":"37","author":"Li","year":"2021","journal-title":"Graphs Comb."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.dam.2022.08.015","article-title":"The 3-path-connectivity of the hypercubes","volume":"322","author":"Zhu","year":"2022","journal-title":"Discret. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/j.dam.2023.06.034","article-title":"3-path-connectivity of Cayley graphs generated by transposition trees","volume":"339","author":"Jin","year":"2023","journal-title":"Discret. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zhu, W.H., Hao, R.X., Feng, Y.Q., and Lee, J. (2023). The 3-path-connectivity of the k-ary n-cube. Appl. Math. Comput., 436.","DOI":"10.1016\/j.amc.2022.127499"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.dam.2024.06.009","article-title":"The 3-path-connectivity of the star graphs","volume":"357","author":"Li","year":"2024","journal-title":"Discret. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Wang, J., and Cheng, D. (2025). The 3-path-connectivity of pancake graphs. J. Supercomput., 81.","DOI":"10.1007\/s11227-025-07558-3"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Cano, A., Camarero, C., Mart\u00ednez, C., and Beivide, R. (2026). A new switch buffer architecture for dragonfly networks. J. Parallel Distrib. Comput., 209.","DOI":"10.1016\/j.jpdc.2025.105199"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.tcs.2022.04.028","article-title":"Connectivity and constructive algorithms of disjoint paths in dragonfly networks","volume":"922","author":"Wu","year":"2022","journal-title":"Theor. Comput. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Guo, H., Hao, R.X., and Wu, J. (2025). 2-edge-Hamilton-connected dragonfly network. J. Parallel Distrib. Comput., 202.","DOI":"10.1016\/j.jpdc.2025.105095"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/76\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,26]],"date-time":"2026-01-26T16:27:59Z","timestamp":1769444879000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/76"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,21]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["axioms15010076"],"URL":"https:\/\/doi.org\/10.3390\/axioms15010076","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,21]]}}}