{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T11:09:27Z","timestamp":1770289767679,"version":"3.49.0"},"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>A well-known result of Tutte says that if $\\Gamma$ is an Abelian group and $G$ is a graph having a nowhere-zero $\\Gamma$-flow, then $G$ has a nowhere-zero $\\Gamma'$-flow for each Abelian group $\\Gamma'$ whose order is at least the order of $\\Gamma$. Jaeger, Linial, Payan, and Tarsi observed that this does not extend to their more general concept of group connectivity. Motivated by this we define $g(k)$ as the least number such that, if $G$ is $\\Gamma$-connected for some Abelian group $\\Gamma$ of order $k$, then $G$ is also $\\Gamma'$-connected for every Abelian group $\\Gamma'$ of order $|\\Gamma'| \\geqslant g(k)$. We prove that $g(k)$ exists and satisfies for infinitely many $k$,\r\n\\begin{align*}(2-o(1)) k &lt; g(k) \\leqslant 8k^3+1.\\end{align*}\r\nThe upper bound holds for all $k$. Analogously, we define $h(k)$ as the least number such that, if $G$ is $\\Gamma$-colorable for some Abelian group $\\Gamma$ of order $k$, then $G$ is also $\\Gamma'$-colorable for every Abelian group $\\Gamma'$ of order $|\\Gamma'| \\geq h(k)$. Then $h(k)$ exists and satisfies for infinitely many $k$,\r\n\\begin{align*}(2-o(1)) k &lt; h(k) &lt; (2+o(1))k \\ln(k).\\end{align*}\r\nThe upper bound (for all $k$) follows from a result of Kr\u00e1l', Pangr\u00e1c, and Voss. The lower bound follows by duality from our lower bound on $g(k)$ as that bound is demonstrated by planar graphs.<\/jats:p>","DOI":"10.37236\/8984","type":"journal-article","created":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:37:29Z","timestamp":1584578249000},"source":"Crossref","is-referenced-by-count":2,"title":["Group Connectivity and Group Coloring: Small Groups versus Large Groups"],"prefix":"10.37236","volume":"27","author":[{"given":"Rikke","family":"Langhede","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carsten","family":"Thomassen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,3,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p49\/8037","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p49\/8037","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:37:29Z","timestamp":1584578249000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,6]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8984","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,6]]},"article-number":"P1.49"}}