{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:45:25Z","timestamp":1759063525731,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We consider $2$-colourings $f : E(G) \\rightarrow \\{ -1 ,1 \\}$ of the edges of a graph $G$ with colours $-1$ and $1$ in $\\mathbb{Z}$. A subgraph $H$ of $G$ is said to be a zero-sum subgraph of $G$ under $f$ if $f(H) := \\sum_{e\\in E(H)} f(e) =0$. We study the following type of questions, in several cases obtaining best possible results: Under which conditions on $|f(G)|$ can we guarantee the existence of a zero-sum spanning tree of $G$? The types of $G$ we consider are complete graphs, $K_3$-free graphs, $d$-trees, and maximal planar graphs. We also answer the question of when any such colouring contains a zero-sum spanning path or a zero-sum spanning tree of diameter at most $3$, showing in passing that the diameter-$3$ condition is best possible. Finally, we give, for $G = K_n$, a sharp bound on $|f(K_n)|$ by which an interesting zero-sum connectivity property is forced, namely that any two vertices are joined by a zero-sum path of length at most $4$.\r\nOne feature of this paper is the proof of an Interpolation Lemma leading to a Master Theorem from which many of the above results follow and which can be of independent interest.<\/jats:p>","DOI":"10.37236\/10289","type":"journal-article","created":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T03:11:19Z","timestamp":1643771479000},"source":"Crossref","is-referenced-by-count":5,"title":["On Zero-Sum Spanning Trees and Zero-Sum Connectivity"],"prefix":"10.37236","volume":"29","author":[{"given":"Yair","family":"Caro","sequence":"first","affiliation":[]},{"given":"Adriana","family":"Hansberg","sequence":"additional","affiliation":[]},{"given":"Josef","family":"Lauri","sequence":"additional","affiliation":[]},{"given":"Christina","family":"Zarb","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2022,1,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p9\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T03:12:30Z","timestamp":1643771550000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i1p9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1,27]]}},"URL":"https:\/\/doi.org\/10.37236\/10289","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,1,28]]},"article-number":"P1.9"}}