The supereulerian width of a graph G, denoted sw(G), is the largest integer s such that for any integer k with \(0 \le k \le s\) , and for any distinct vertices \(u, v \in V(G)\) , G has a spanning subgraph H consisting of k-edge-disjoint (u, v)-trails. It is known that for any graph G, \(\kappa '(G) \ge sw(G)\) . As deciding if \(sw(G) \ge 2\) is NP-complete, Li et al. in 2016 posed an open problem asking to determine sw(G) when \(\kappa '(G)\) is given. Former results in 1988 of Catlin on supereulerian graphs and in 2009 of Lai et al. imply that every 4-edge-connected graph G satisfies \(sw(G) \ge 2\) . We proved that every 4-edge-connected graph with diameter at most 4 satisfies \(sw(G) \ge 3\) ; and every 3-edge-connected claw-free graph with diameter at most 3 satisfies \(sw(G) \ge 3\) .
