{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:42:02Z","timestamp":1753890122521,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Graph Theory","license":[{"start":{"date-parts":[[2018,1,17]],"date-time":"2018-01-17T00:00:00Z","timestamp":1516147200000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2018,1,17]],"date-time":"2018-01-17T00:00:00Z","timestamp":1516147200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2018,1,17]],"date-time":"2018-01-17T00:00:00Z","timestamp":1516147200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>We introduce a natural variant of the parallel chip-firing game, called the diffusion game. Chips are initially assigned to vertices of a graph. At every step, all vertices simultaneously send one chip to each neighbour with fewer chips. As the dynamics of the parallel chip-firing game occur on a finite set the process is inherently periodic. However the diffusion game is not obviously periodic: even if $2|E(G)|$ chips are assigned to vertices of graph G, there may exist time steps where some vertices have a negative number of chips. We investigate the process, prove periodicity for a number of graph classes, and pose some questions for future research.<\/jats:p><jats:p>Comment: 18 pages, 3 figures<\/jats:p>","DOI":"10.23638\/dmtcs-20-1-4","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:39:35Z","timestamp":1743698375000},"source":"Crossref","is-referenced-by-count":0,"title":["A Variation on Chip-Firing: the diffusion game"],"prefix":"10.23638","volume":"Vol. 20 no. 1","author":[{"given":"C.","family":"Duffy","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T. F.","family":"Lidbetter","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M. E.","family":"Messinger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R. J.","family":"Nowakowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2018,1,17]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1609.05792v5","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1609.05792v5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:39:35Z","timestamp":1743698375000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/2039"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,17]]},"references-count":0,"journal-issue":{"issue":"Graph Theory","published-online":{"date-parts":[[2018,1,17]]}},"URL":"https:\/\/doi.org\/10.23638\/dmtcs-20-1-4","relation":{"has-preprint":[{"id-type":"arxiv","id":"1609.05792v3","asserted-by":"subject"},{"id-type":"arxiv","id":"1609.05792v2","asserted-by":"subject"},{"id-type":"arxiv","id":"1609.05792v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1609.05792","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1609.05792","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2018,1,17]]},"article-number":"2039"}}