Let G be the Cartesian product of a regular tree T and a finite connected transitive graph H. It is shown in [3] that the Free Uniform Spanning Forest (FSF) of this graph may not be connected, but the dependence of this connectedness on H remains somewhat mysterious. We study the case when a positive weight w is put on the edges of the H-copies in G, and conjecture that the connectedness of the FSF exhibits a phase transition. For large enough w we show that the FSF is connected, while for a large family of H and T , the FSF is disconnected when w is small (relying on [3]). Finally, we prove that when H is the graph of one edge, then for any w, the FSF is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.

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Given a graph G with only even degrees let ε(G) be the number of Eulerian orientations and let h(G) denote the number of half graphs, that is, subgraphs F such that d F (v) = d G (v)/2 for each vertex v. Recently, M. Borbényi and P. Csikvári proved that ε(G) ≥ h(G) holds true for all Eulerian graphs with equality if and and only if G is bipartite. In this paper we give a simple new proof of this fact, and we give identities and inequalities for the number of Eulerian orientations and half graphs of a 2-cover of a graph G.

Given a graph $G$ with only even degrees, let $\varepsilon(G)$ denote the number of Eulerian orientations, and let $h(G)$ denote the number of half graphs, that is, subgraphs $F$ such that $d_F(v)=d_G(v)/2$ for each vertex $v$. Recently, Borbényi and Csikvári proved that  $\varepsilon(G)\geq h(G)$ holds true for all Eulerian graphs, with equality if and only if $G$ is bipartite. In this paper we give a simple new proof of this fact, and we give identities and inequalities for the number of Eulerian orientations and half graphs of a $2$-cover of a graph $G$.