{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T20:12:58Z","timestamp":1772136778338,"version":"3.50.1"},"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>Consider a multigraph $G$ whose edges are colored from $[q]$ ($q$-colored graph) and $\\alpha=(\\alpha_1,\\ldots,\\alpha_{q}) \\in \\mathbb{N}^{q}$ (color-constraint). A subgraph $H$ of $G$ is called $\\alpha$-colored if $H$ has exactly $\\alpha_i$ edges of color $i$ for each $i \\in[q]$. In this paper, we focus on $\\alpha$-colored arborescences (spanning out-trees) in $q$-colored multidigraphs. We study the decision, counting and search versions of this problem. It is known that the decision and search problems are polynomial-time solvable when $q=2$ [Barahona and Pulleyblank, Discret. Appl. Math. 1987] and that the decision problem is NP-complete when $q$ is arbitrary [Ardra et al., arXiv 2024]. However the complexity status of the problem for fixed $q$ was open for $q &gt; 2$.\r\nWe solve this problem using an algebraic approach. Given a $q$-colored digraph $G$ and a vertex $s$ in $G$, we construct a symbolic matrix in $q-1$ indeterminates such that the number of $\\alpha$-colored arborescences in $G$ rooted at $s$ for all color-constraints $\\alpha \\in \\mathbb{N}^q$ can be read from its determinant polynomial. This result extends Tutte's matrix-tree theorem and gives a polynomial-time algorithm for the counting and decision problems for fixed $q$. We use it to design an algorithm that finds an $\\alpha$-colored arborescence when one exists. We also study the weighted variant of the problem and give a polynomial-time algorithm (when $q$ is fixed and weights are polynomially bounded) which finds a minimum weight solution.<\/jats:p>","DOI":"10.37236\/14453","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:25Z","timestamp":1772135005000},"source":"Crossref","is-referenced-by-count":0,"title":["Color-Constrained Arborescences in Edge-Colored Digraphs"],"prefix":"10.37236","volume":"33","author":[{"family":"P. S. Ardra","sequence":"first","affiliation":[]},{"given":"Jasine","family":"Babu","sequence":"additional","affiliation":[]},{"family":"R. Krithika","sequence":"additional","affiliation":[]},{"family":"Deepak Rajendraprasad","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,2,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p34\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:26Z","timestamp":1772135006000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,27]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/14453","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,27]]},"article-number":"P1.34"}}