Edge-disjoint spanning trees and the number of maximum state circles of a graph

Motivated by the connection with the genus of unoriented alternating links, Jin et al. (Acta Math Appl Sin Engl Ser, 2015) introduced the number of maximum state circles of a plane graph G, denoted by \(s_{\max }(G)\), and proved that \(s_{\max }(G)=\max \{e(H)+2c(H)-v(H)|\) H is a spanning subgraph of \(G\}\), where e(H), c(H) and v(H) denote the size, the number of connected components and the order of H, respectively. In this paper, we show that for any (not necessarily planar) graph G, \(s_{\max }(G)\) can be achieved by the spanning subgraph H of G whose each connected component is a maximal subgraph of G with two edge-disjoint spanning trees. Such a spanning subgraph is proved to be unique and we present a polynomial-time algorithm to find such a spanning subgraph for any graph G.