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Register a new userMany edge-disjoint rainbow spanning trees in general graphs
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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.
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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 maximum number of colors in an edge-coloring of $G$ containing no $t$ edge-disjoint rainbow spanning trees. For any vertex partition $P$, let $E(P,G)$ be the set of non-crossing edges in $G$ with respect to $P$. In this paper, we determine $r(G,t)$ for all host graphs $G$: $r(G,t)=|E(G)|$ if there exists a partition $P_0$ with $|E(G)|-|E(P_0,G)|<t(|P_0|-1)$; and $r(G,t)=max_{Pcolon |P|geq 3} {|E(P,G)|+t(|P|-2)}$ otherwise. As a corollary, we determine $r(K_{p,q},t)$ for all values of $p,q, t$, improving a result of Jia, Lu and Zhang.
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 $G$. Such a question clearly needs restrictions on the colorings to be meaningful. For edge-colorings using $n-1$ colors and without rainbow cycles, known in the literature as JL-colorings, there turns out to be a particularly nice way of counting the rainbow spanning trees and we solve this problem completely for JL-colored complete graphs $K_n$ and complete bipartite graphs $K_{n,m}$. In both cases, we find tight upper and lower bounds; the lower bound for $K_n$, in particular, proves to have an unexpectedly chaotic and interesting behavior. We further investigate this question for JL-colorings of general graphs and prove several results including characterizing graphs which have JL-colorings achieving the lowest possible number of rainbow spanning trees. We establish other results for general $n-1$ colorings, including providing an analogue of Kirchoffs matrix tree theorem which yields a way of counting rainbow spanning trees in a general graph $G$.
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 result asserts that a uniform colouring of $mathbb{G}(n,omega(1)/n)$, using a palette of size $n$, a.a.s. admits a rainbow copy of any given bounded-degree tree on at most $(1-varepsilon)n$ vertices, where $varepsilon > 0$ is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon, Krivelevich, and Sudakov pertaining to the embedding of bounded-degree almost-spanning prescribed trees in $mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$. Given an $n$-vertex graph $G$ with minimum degree at least $delta n$, where $delta > 0$ is fixed, we use our aforementioned result in order to prove that a uniform colouring of the randomly perturbed graph $G cup mathbb{G}(n,omega(1)/n)$, using $(1+alpha)n$ colours, where $alpha > 0$ is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree {sl spanning} tree. This can be viewed as a rainbow variant of a result by Krivelevich, Kwan, and Sudakov who proved that $G cup mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$, a.a.s. admits a copy of any given bounded-degree spanning tree. Finally, and with $G$ as above, we prove that a uniform colouring of $G cup mathbb{G}(n,omega(n^{-2}))$ using $n-1$ colours a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.
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.
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 maximum degree at most $cn/log n$. We prove the corresponding result for directed graphs. That is, for each $alpha>0$, there is some $c>0$ and $n_0$ such that, if $ngeq n_0$, then every $n$-vertex directed graph with minimum semi-degree at least $(1/2+alpha)n$ contains a copy of every $n$-vertex oriented tree whose underlying maximum degree is at most $cn/log n$. As with Komlos, Sarkozy and Szemeredis theorem, this is tight up to the value of $c$. Our result improves a recent result of Mycroft and Naia, which requires the oriented trees to have underlying maximum degree at most $Delta$, for any constant $Delta$ and sufficiently large $n$. In contrast to the previous work on spanning trees in dense directed or undirected graphs, our methods do not use Szemeredis regularity lemma.
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