{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:30Z","timestamp":1753893810439,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Cartwright (2015) introduced the notion of a weak tropical complex in order to generalize the theory of divisors on graphs from Baker and Norine (2007). A weak tropical complex $\\Gamma$ is a $\\Delta$-complex equipped with algebraic data that allows it to be viewed as the dual complex to a certain kind of degeneration over a discrete valuation ring. Every graph has a unique tropical complex structure (which is the same structure studied by Baker and Norine) in which divisors correspond to states in the chip-firing game on that graph. Let $G$ and $H$ be graphs, and let $\\Gamma$ be a triangulation of $G\\times H$ obtained by adding in one diagonal of each resulting square. There is a particular weak tropical complex structure on $\\Gamma$ that Cartwright conjectured was closely related to the weak tropical complex structures on $G$ and $H$. The main result of this paper is a proof of Cartwright's conjecture. In preparation, we discuss some basic properties of tropical complexes, along with some properties specific to the product-of-graphs case.<\/jats:p>","DOI":"10.37236\/5533","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:44:41Z","timestamp":1578653081000},"source":"Crossref","is-referenced-by-count":0,"title":["A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group"],"prefix":"10.37236","volume":"24","author":[{"given":"Alexander","family":"Lazar","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,10,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p14\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p14\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:45:01Z","timestamp":1579218301000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i4p14"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,20]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,10,5]]}},"URL":"https:\/\/doi.org\/10.37236\/5533","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,10,20]]},"article-number":"P4.14"}}