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A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph $K_{2n}$, there exists a decomposition of $K_{2n}$ into isomorphic rainbow spanning trees. This settles conjectures of Brualdi--Hollingsworth (from 1996) and Constantine (from 2002) for large graphs.
A rainbow spanning tree in an edge-colored graph is a spanning tree in which each edge is a different color. Carraher, Hartke, and Horn showed that for $n$ and $C$ large enough, if $G$ is an edge-colored copy of $K_n$ in which each color class has si
A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives a distinct color. In this paper we consider the natural extremal problem of maximizing and minimizing the number of rainbow spanning trees in a graph $G$. Su
An edge-colored graph $G$ is called textit{rainbow} if every edge of $G$ receives a different color. Given any host graph $G$, the textit{anti-Ramsey} number of $t$ edge-disjoint rainbow spanning trees in $G$, denoted by $r(G,t)$, is defined as the m
Given a collection of graphs $mathbf{G}=(G_1, ldots, G_m)$ with the same vertex set, an $m$-edge graph $Hsubset cup_{iin [m]}G_i$ is a transversal if there is a bijection $phi:E(H)to [m]$ such that $ein E(G_{phi(e)})$ for each $ein E(H)$. We give asy
We study the following rainbow version of subgraph containment problems in a family of (hyper)graphs, which generalizes the classical subgraph containment problems in a single host graph. For a collection $textbf{G}={G_1, G_2,ldots, G_{m}}$ of not ne