Up-embeddability of graphs with new degree-sum

Let G be a k ( k ≤ 2)-edge connected simple graph with minimal degree ≥ 3 and girth g,r = ⌊g - 1/2⌋ . For any edge uv ∈ E ( G ), if d_G( u ) + d_G( v ) > 2v( G ) - 2( k + 1)( g - 2r)/( k + 1)( 2^r - 1)( g - 2r) + 2( g - 2r - 1), then G is up-embeddable. Furthermore, similar results for 3-edge connected simple graphs are also obtained.