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−12⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g,r = \left\lfloor {\frac{{g - 1}}{2}} \right\rfloor $$\end{document} . For any edge uv ∈ E(G), if dG(u)+dG(v)>2v(G)−2(k+1)(g−2r)(k+1)(2r−1)(g−2r)+2(g−2r−1),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d_G}\left( u \right) + {d_G}\left( v \right) > \frac{{2v\left( G \right) - 2\left( {k + 1} \right)\left( {g - 2r} \right)}}{{\left( {k + 1} \right)\left( {{2^r} - 1} \right)\left( {g - 2r} \right)}} + 2\left( {g - 2r - 1} \right),$$\end{document} then G is up-embeddable. Furthermore, similar results for 3-edge connected simple graphs are also obtained.