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Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>This paper studies the fundamental relations among integer flows, modulo orientations, integer-valued and real-valued circular flows, and monotonicity of flows in signed graphs. A (signed) graph is modulo-$(2p+1)$-orientable if it has an orientation such that the indegree is congruent to the outdegree modulo $2p+1$ at each vertex. An integer-valued $\\frac{2p+1}{p}$-flow is a flow taking integer values in $\\{\\pm p, \\pm (p+1)\\}$. Extending a fundamental result of Jaeger to signed graphs, we show that a bridgeless signed graph is modulo-$(2p+1)$-orientable if and only if it admits an integer-valued $\\frac{2p+1}{p}$-flow. It was conjectured by Raspaud and Zhu that, for any signed graph, the admission of a circular $r$-flow implies the admission of an integer-valued $\\lceil r \\rceil$-flow. Although this conjecture has been disproved in general, it is confirmed in this paper for bridgeless signed graphs if $r=\\frac{2p+1}{p}$ and $p \\geq 3$.<\/jats:p>","DOI":"10.1137\/20m1317141","type":"journal-article","created":{"date-parts":[[2021,3,11]],"date-time":"2021-03-11T10:19:02Z","timestamp":1615457942000},"page":"376-391","source":"Crossref","is-referenced-by-count":2,"title":["Integer Flows and Modulo Orientations of Signed Graphs"],"prefix":"10.1137","volume":"35","author":[{"given":"Miaomiao","family":"Han","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4771-2094","authenticated-orcid":true,"given":"Jiaao","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rong","family":"Luo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yongtang","family":"Shi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5583-4481","authenticated-orcid":true,"given":"Cun-Quan","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,3,11]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01214296"},{"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":"crossref","unstructured":"R. Diestel,\n                      Graph Theory\n                      , 4th ed., Springer, Heidelberg, 2010.","DOI":"10.1007\/978-3-642-14279-6"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(83)90041-2"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/17M1126072"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"M. DeVos, J. Li, Y. Lu, R. Luo, C.Q. Zhang, and Z. Zhang,\n                      Flows on flow-admissible signed graphs\n                      , J. Combin. Theory Ser. B, available online (2020),https:\/\/doi.org\/10.1016\/j.jctb.2020.04.008.","DOI":"10.1016\/j.jctb.2020.04.008"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01387754"},{"key":"atypb8","unstructured":"H. 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Wilson, eds., Academic Press, San Diego, CA, 1988, pp. 71-95."},{"key":"atypb14","first-page":"877","volume":"88","author":"Kompi\u0161ov\u00e1 A.","year":"2019","journal-title":"S.)"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190090111"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(86)90025-0"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2020.06.001"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"L.M. Lov\u00e1sz, C. Thomassen, Y. Wu, and C.Q. Zhang,\n                      Nowhere-zero \\textup3-flows and modulo $k$-orientations\n                      , J. Combin. Theory Ser. 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