Supereulerian width of dense graphs

For a graph G, the supereulerian widthμ′(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)≥n4−1, then μ′(G)≥2. In this paper, we show that for any real numbers a,b with 00, 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 uv⁄∈E(G), max{dG(u),dG(v)}≥an+b, then either μ′(G)≥s+1 or G is contractible to a member in F. When a=14,b=−32, we show that if n is sufficiently large, K3,3 is the only obstacle for a 3-edge-connected graph G to satisfy μ′(G)≥3.