{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T20:04:39Z","timestamp":1768075479065,"version":"3.49.0"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Graph Theory","license":[{"start":{"date-parts":[[2018,12,17]],"date-time":"2018-12-17T00:00:00Z","timestamp":1545004800000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2018,12,17]],"date-time":"2018-12-17T00:00:00Z","timestamp":1545004800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2018,12,17]],"date-time":"2018-12-17T00:00:00Z","timestamp":1545004800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>A $\\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \\ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\\textit{ sigma number}$ of a graph $G$, denoted by $\\sigma(G)$, is the minimum number $k$ such that $ G $ has a sigma partitioning $P_1, \\ldots, P_k$. Also, a $\\textit{ lucky labeling}$ of a graph $G$ is a function $ \\ell :V(G) \\rightarrow \\mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \\sum_{w \\sim v}\\ell(w)\\neq \\sum_{w \\sim u}\\ell(w) $ ($ x \\sim y $ means that $ x $ and $y$ are adjacent). The $\\textit{ lucky number}$ of $ G $, denoted by $\\eta(G)$, is the minimum number $k $ such that $ G $ has a lucky labeling $ \\ell :V(G) \\rightarrow \\mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $ \\mathbf{NP} $-complete to decide whether $ \\eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph.<\/jats:p>","DOI":"10.23638\/dmtcs-20-2-19","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:40:08Z","timestamp":1743698408000},"source":"Crossref","is-referenced-by-count":1,"title":["Sigma Partitioning: Complexity and Random Graphs"],"prefix":"10.23638","volume":"vol. 20 no. 2","author":[{"given":"Ali","family":"Dehghan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammad-Reza","family":"Sadeghi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arash","family":"Ahadi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2018,12,17]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1403.6288v7","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1403.6288v7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:40:09Z","timestamp":1743698409000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/1534"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,12,17]]},"references-count":0,"journal-issue":{"issue":"Graph Theory","published-online":{"date-parts":[[2018,12,17]]}},"URL":"https:\/\/doi.org\/10.23638\/dmtcs-20-2-19","relation":{"has-preprint":[{"id-type":"arxiv","id":"1403.6288v4","asserted-by":"subject"},{"id-type":"arxiv","id":"1403.6288v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1403.6288","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1403.6288","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"value":"1365-8050","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,12,17]]},"article-number":"1534"}}