{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:04:34Z","timestamp":1787321074378,"version":"3.56.0"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2014,1]]},"abstract":"<jats:p>An orientation of a graph $G$ is a mod (2p+1)-orientation if, under this orientation, the net out-degree at every vertex is congruent to zero mod 2p+1. If, for any function $b: V(G) \\rightarrow \\mathbb Z_{2p+1}$ satisfying $\\sum_{v \\in V(G)} b(v) \\equiv 0$ (mod 2p+1), $G$ always has an orientation $D$ such that the net out-degree at every vertex $v$ is congruent to $b(v)$ mod 2p+1, then $G$ is strongly $\\mathbb Z_{2p+1}$-connected. The graph $G'$ obtained from $G$ by contracting all nontrivial subgraphs that are strongly $\\mathbb Z_{2s+1}$-connected is called the $\\mathbb Z_{2s+1}$-reduction of $G$. Motivated by a minimum degree condition of Barat and Thomassen [J. Graph Theory, 52 (2006), pp. 135--146], and by the Ore conditions of Fan and Zhou [SIAM J. Discrete Math., 22 (2008), pp. 288--294] and of Luo et al. [European J. Combin., 29 (2008), pp. 1587--1595] on $\\mathbb Z_3$-connected graphs, we prove that for a simple graph $G$ on $n$ vertices, and for any integers $s &gt; 0$ and real numbers $\\alpha, \\beta$ with $0 &lt; \\alpha &lt; 1$, if for any nonadjacent vertices $u, v \\in V(G)$, $d_G(u) + d_G(v) \\ge \\alpha n + \\beta$, then there exists a finite family ${\\cal {F}}(\\alpha,s)$ of nonstrongly $\\mathbb Z_{2s+1}$-connected graphs such that either $G$ is strongly $\\mathbb Z_{2s+1}$-connected or the $\\mathbb Z_{2s+1}$-reduction of $G$ is in ${\\cal {F}}(\\alpha,s)$.<\/jats:p>","DOI":"10.1137\/130920435","type":"journal-article","created":{"date-parts":[[2014,10,14]],"date-time":"2014-10-14T13:29:23Z","timestamp":1413293363000},"page":"1820-1827","source":"Crossref","is-referenced-by-count":2,"title":["On Mod $(2s+1)$-Orientations of Graphs"],"prefix":"10.1137","volume":"28","author":[{"given":"Ping","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong-Jian","family":"Lai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2014,10,14]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20149"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"J. A. Bondy and U. S. R. Murty,\n                      Graph Theory\n                      , Springer, New York, 2008.","DOI":"10.1007\/978-1-84628-970-5"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(95)00150-U"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(00)00071-6"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/060677744"},{"key":"atypb7","first-page":"391","author":"Jaeger F.","year":"1984","journal-title":"Amsterdam"},{"key":"atypb8","first-page":"91","author":"Jaeger F.","year":"1988","journal-title":"New York"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(92)90016-Q"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.2001.2054"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/060676945"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2014.03.017"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2008.08.001"},{"key":"atypb14","unstructured":"Y. T. Liang,\n                      Cycles, Disjoint Spanning Trees, and Orientation of Graphs\n                      , Ph.D. dissertation, West Virginia University, Morgantown, WV, 2012."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2013.06.003"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2007.11.014"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2011.09.003"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1954-010-9"},{"key":"atypb19","unstructured":"Y. Wu,\n                      Integer Flows and Modulo Orientations\n                      , Ph.D. dissertation, West Virginia University, Morgantown, WV, 2012."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/130920435","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:07:42Z","timestamp":1787317662000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/130920435"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,1]]},"references-count":18,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["10.1137\/130920435"],"URL":"https:\/\/doi.org\/10.1137\/130920435","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,1]]}}}