On spanning tree edge denpendences of graphs

Abstract: Let $\tau(G)$ and $\tau_G(e)$ be the number of spanning trees of a connected graph $G$ and the number of spanning trees of $G$ containing edge $e$. The ratio $d_{G}(e)=\tau_{G}(e)/\tau(G)$ is called the spanning tree edge density of $e$, or simply density of $e$. The maximum density $\mbox{dep}(G)=\max\limits_{e\in E(G)}d_{G}(e)$ is called the spanning tree edge dependence of $G$, or simply dependence of $G$. Given a rational number $p/q\in (0,1)$, if there exists a graph $G$ and an edge $e\in E(G)$ such that $d_{G}(e)=p/q$, then we say the density $p/q$ is constructible. More specially, if there exists a graph $G$ such that $\mbox{dep}(G)=p/q$, then we say the dependence $p/q$ is constructible. In 2002, Ferrara, Gould, and Suffel raised the open problem of which rational densities and dependences are constructible. In 2016, Kahl provided constructions that show all rational densities and dependences are constructible. Moreover, He showed that all rational densities are constructible even if $G$ is restricted to bipartite graphs or planar graphs. He thus conjectured that all rational dependences are also constructible even if $G$ is restricted to bipartite graphs (Conjecture 1), or planar graphs (Conjecture 2). In this paper, by combinatorial and electric network approach, firstly, we show that all rational dependences are constructible via bipartite graphs, which confirms the first conjecture of Kahl. Secondly, we show that all rational dependences are constructible for planar multigraphs, which confirms Kahl's second conjecture for planar multigraphs. However, for (simple) planar graphs, we disprove the second conjecture of Kahl by showing that the dependence of any planar graph is larger than $\frac{1}{3}$. On the other hand, we construct a family of planar graphs that show all rational dependences $p/q>\frac{1}{2}$ are constructible via planar graphs.