For a graph G , the supereulerian width μ ′ ( G ) of a graph G is the largest integer s such that G has a spanning ( k ; u , v ) -trail-system, for any integer k with 1 ≤ k ≤ s , and for any u , v ∈ V ( G ) with u ≠ v . Thus μ ′ ( G ) ≥ 2 implies that G is supereulerian, and so graphs with higher supereulerian width are natural generalizations of supereulerian graphs. Settling an open problem of Bauer, Catlin (1988) proved that if a simple graph G on n ≥ 17 vertices satisfy δ ( G ) ≥ n 4 − 1 , then μ ′ ( G ) ≥ 2 . In this paper, we show that for any real numbers a , b with 0 < a < 1 and any integer s > 0 , there exists a finite graph family F = F ( a , b , s ) such that for a simple graph G with n = | V ( G ) | , if for any u , v ∈ V ( G ) with u v ⁄ ∈ E ( G ) , max { d G ( u ) , d G ( v ) } ≥ a n + b , then either μ ′ ( G ) ≥ s + 1 or G is contractible to a member in F . When a = 1 4 , b = − 3 2 , we show that if n is sufficiently large, K 3 , 3 is the only obstacle for a 3-edge-connected graph G to satisfy μ ′ ( G ) ≥ 3 .