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The goal is to find a fair, i.e.,\n                    <jats:italic>envy-free up to any item<\/jats:italic>\n                    (\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {EFX}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    ) allocation. This model has recently been introduced by [22] where they show that\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {EFX}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    allocations always exist on simple graphs for\n                    <jats:italic>monotone<\/jats:italic>\n                    valuations, i.e., where any two agents can share at most one edge (item). A natural question arises as to what happens when we go beyond simple graphs and study various classes of multi-graphs? We answer the above question affirmatively for the valuation class of\n                    <jats:italic>bipartite multi-graphs<\/jats:italic>\n                    and\n                    <jats:italic>multi-cycles<\/jats:italic>\n                    . The main contribution of this work is to establish the existence of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {EFX}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    allocations on bipartite multi-graphs for\n                    <jats:italic>monotone<\/jats:italic>\n                    valuations and on multi-cycles for\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {MMS}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    <jats:italic>-feasible<\/jats:italic>\n                    valuations. We also present pseudo-polynomial time algorithms to compute\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {EFX}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    allocations for the above settings. Furthermore, we show that for bipartite multi-graphs with\n                    <jats:italic>cancelable<\/jats:italic>\n                    valuations,\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {EFX}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    allocations can be computed in polynomial time. We thus deepen the understanding of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {EFX}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    allocations by expanding the spectrum of settings in which they are guaranteed to exist for an arbitrary number of agents. Next, we study\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {EFX}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    <jats:italic>orientations<\/jats:italic>\n                    (allocations where every item is assigned to one of its two endpoint agents) and provide a complete characterization of their existence on bipartite multi-graphs in terms of two key parameters\u2014the number of edges shared between any two agents and the diameter of the graph. Finally, we prove that it is\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {NP}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -complete to determine whether a given fair division instance on a bipartite multi-graph admits an\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textsf {EFX}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    orientation, even with a constant number of agents.\n                  <\/jats:p>","DOI":"10.1007\/s10458-026-09754-8","type":"journal-article","created":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T08:15:47Z","timestamp":1780388147000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["$$\\textsf {EFX}$$ allocations and orientations on bipartite multi-graphs: a complete picture"],"prefix":"10.1007","volume":"40","author":[{"given":"Mahyar","family":"Afshinmehr","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alireza","family":"Danaei","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mehrafarin","family":"Kazemi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kurt","family":"Mehlhorn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nidhi","family":"Rathi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,2]]},"reference":[{"key":"9754_CR1","unstructured":"Afshinmehr, M.,\u00a0Ansaripour, M.,\u00a0Danaei, A., &\u00a0Mehlhorn, K. 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