{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T11:30:22Z","timestamp":1768735822082,"version":"3.49.0"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,9]],"date-time":"2025-01-09T00:00:00Z","timestamp":1736380800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science and Technology Council, Taiwan","award":["NSTC 111-2115-M-260-001-"],"award-info":[{"award-number":["NSTC 111-2115-M-260-001-"]}]},{"name":"National Science and Technology Council, Taiwan","award":["NSTC 112-2115-M-260-001-MY2"],"award-info":[{"award-number":["NSTC 112-2115-M-260-001-MY2"]}]},{"name":"National Science and Technology Council, Taiwan","award":["NSTC 111-2115-M-260-001-"],"award-info":[{"award-number":["NSTC 111-2115-M-260-001-"]}]},{"name":"National Science and Technology Council of Taiwan","award":["NSTC 111-2115-M-260-001-"],"award-info":[{"award-number":["NSTC 111-2115-M-260-001-"]}]},{"name":"National Science and Technology Council of Taiwan","award":["NSTC 112-2115-M-260-001-MY2"],"award-info":[{"award-number":["NSTC 112-2115-M-260-001-MY2"]}]},{"name":"National Science and Technology Council of Taiwan","award":["NSTC 111-2115-M-260-001-"],"award-info":[{"award-number":["NSTC 111-2115-M-260-001-"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Finding a Hamiltonian cycle in a graph G = (V, E) is a well-known problem. The challenge of finding a Hamiltonian cycle that avoids these faults when faulty vertices or edges are present has been extensively studied. When the edge set of G is partitioned into k dimensions, the problem of dimension-balanced Hamiltonian cycles arises, where the Hamiltonian cycle uses approximately the same number of edges from each dimension (differing by at most one). This paper studies whether a dimension-balanced Hamiltonian cycle (DBH) exists in toroidal mesh graphs Tm,n when a single vertex or edge is faulty, called the one-fault DBH problem. We establish that Tm,n is one-fault DBH, except in the following cases: (1) both m and n are even; (2) one of m and n is 3, while the other satisfies mod 4 = 3 and is greater than 6; (3) one of m and n is odd, while the other satisfies mod 4 = 2. Additionally, this paper resolves a conjecture from prior literature, thereby providing a complete solution to the DBP problem on Tm,n.<\/jats:p>","DOI":"10.3390\/sym17010093","type":"journal-article","created":{"date-parts":[[2025,1,9]],"date-time":"2025-01-09T07:59:42Z","timestamp":1736409582000},"page":"93","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The One-Fault Dimension-Balanced Hamiltonian Problem in Toroidal Mesh Graphs"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3654-2560","authenticated-orcid":false,"given":"Justie Su-Tzu","family":"Juan","sequence":"first","affiliation":[{"name":"Department of Computer Science and Information Engineering, National Chi Nan University, Puli, Nantou 545, Taiwan"}]},{"given":"Hao-Cheng","family":"Ciou","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Information Engineering, National Chi Nan University, Puli, Nantou 545, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-5699-3974","authenticated-orcid":false,"given":"Meng-Jyun","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Information Engineering, National Chi Nan University, Puli, Nantou 545, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6337","DOI":"10.1016\/j.disc.2008.11.024","article-title":"Hamiltonicity and pancyclicity of cartesian products of graphs","volume":"309","author":"Cada","year":"2009","journal-title":"Discret. 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