On two conjectures concerning spanning tree edge dependences of graphs

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Let τ(G) and τG(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 dG(e)=τG(e)/τ(G) is called the spanning tree edge density, or simply density, of e. The maximum density dep(G)=maxe∈E(G)dG(e) is called the spanning tree edge dependence, or simply dependence, of G. In 2016, Kahl made the following two conjectures. (C1) Let p,q be positive integers, p<q. There exists some function f(p,q) such that, if G is the bipartite construction as given in Theorem 3.5 (Theorem 2.1 in the present paper), then ti≥f(p,q) for all 2≤i≤n implies that dep(G)=p/q. (C2) Let p,q be positive integers, p<q. There exists a planar graph G such that dep(G)=p/q. In this paper, by using combinatorial and electrical network approaches, firstly, we show (C1) is true. Secondly, we disprove (C2) by showing that for any planar graph G, dep(G)>13. On the other hand, by construction families of planar graphs, we show that, for p/q>12, there do exist a planar graph G such that dep(G)=p/q. Furthermore, we prove that the second conjecture holds for planar multigraphs.