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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 size at most $n/2$, then $G$ has at least $lfloor n/(Clog n)rfloor$ edge-disjoint rainbow spanning trees. Here we strengthen this result by showing that if $G$ is any edge-colored graph with $n$ vertices in which each color appears on at most $deltacdotlambda_1/2$ edges, where $deltageq Clog n$ for $n$ and $C$ sufficiently large and $lambda_1$ is the second-smallest eigenvalue of the normalized Laplacian matrix of $G$, then $G$ contains at least $leftlfloorfrac{deltacdotlambda_1}{Clog n}rightrfloor$ edge-disjoint rainbow spanning trees.
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
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
We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette. Our first re
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 is
In 2001, Komlos, Sarkozy and Szemeredi proved that, for each $alpha>0$, there is some $c>0$ and $n_0$ such that, if $ngeq n_0$, then every $n$-vertex graph with minimum degree at least $(1/2+alpha)n$ contains a copy of every $n$-vertex tree with maxi