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Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well-known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.
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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 asymptotically-tight minimum degree conditions for a graph collection on an $n$-vertex set to have a transversal which is a copy of a graph $H$, when $H$ is an $n$-vertex graph which is an $F$-factor or a tree with maximum degree $o(n/log n)$.
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.
Random spanning trees of a graph $G$ are governed by a corresponding probability mass distribution (or law), $mu$, defined on the set of all spanning trees of $G$. This paper addresses the problem of choosing $mu$ in order to utilize the edges as fairly as possible. This turns out to be equivalent to minimizing, with respect to $mu$, the expected overlap of two independent random spanning trees sampled with law $mu$. In the process, we introduce the notion of homogeneous graphs. These are graphs for which it is possible to choose a random spanning tree so that all edges have equal usage probability. The main result is a deflation process that identifies a hierarchical structure of arbitrary graphs in terms of homogeneous subgraphs, which we call homogeneous cores. A key tool in the analysis is the spanning tree modulus, for which there exists an algorithm based on minimum spanning tree algorithms, such as Kruskals or Prims.
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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