{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:46Z","timestamp":1753893826496,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G$ be a multigraph. A subset $F$ of $E(G)$ is an edge cover of $G$ if every vertex of $G$ is incident to an edge of $F$. The cover index, $\\xi(G)$, is the largest number of edge covers into which the edges of $G$ can be partitioned. Clearly $\\xi(G) \\le \\delta(G)$, the minimum degree of $G$. For $U\\subseteq V(G)$, denote by $E^+(U)$ the set of edges incident to a vertex of $U$. When $|U|$ is odd, to cover all the vertices of $U$, any edge cover needs to contain at least $(|U|+1)\/2$ edges from $E^+(U)$, indicating $ \\xi(G) \\le |E^+(U)|\/ ((|U|+1)\/2)$. Let $\\rho_c(G)$, the co-density of $G$, be defined as the minimum of $|E^+(U)|\/((|U|+1)\/2)$ ranging over all $U\\subseteq V(G)$, where $|U| \\ge 3$ and $|U|$ is odd. Then $\\rho_c(G)$ provides another upper bound on $\\xi(G)$. Thus $\\xi(G) \\le \\min\\{\\delta(G), \\lfloor \\rho_c(G) \\rfloor \\}$. For a lower bound on $\\xi(G)$, in 1978, Gupta conjectured that $\\xi(G) \\ge \\min\\{\\delta(G)-1, \\lfloor \\rho_c(G) \\rfloor \\}$. Gupta himself verified the conjecture for simple graphs, and Cao et al. recently verified this conjecture when $\\rho_c(G)$ is not an integer. In this paper, we confirm the conjecture when the maximum multiplicity of $G$ is at most two or $ \\min\\{\\delta(G)-1, \\lfloor \\rho_c(G) \\rfloor \\} \\le 6$. The proof relies on a newly established result on edge colorings. The result holds independent interest and has the potential to significantly contribute towards resolving the conjecture entirely.<\/jats:p>","DOI":"10.37236\/13163","type":"journal-article","created":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T19:56:24Z","timestamp":1747166184000},"source":"Crossref","is-referenced-by-count":0,"title":["Edge Cover Through Edge Coloring"],"prefix":"10.37236","volume":"32","author":[{"given":"Guantao","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Songling","family":"Shan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,5,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p28\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p28\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T19:56:24Z","timestamp":1747166184000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i2p28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,13]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,4,11]]}},"URL":"https:\/\/doi.org\/10.37236\/13163","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,5,13]]},"article-number":"P2.28"}}