Variations on a theorem of Petersen

For an (r ? 2)-edge-connected graphG (r ≥ 3) for orderp containing at mostk edge cut sets of cardinalityr ? 2 and for an integerl with 0 ≤l ≤ ?p/2?, it is shown that (1) ifp is even, 0 ≤k ≤ r(l + 1) ? 1, and $$\mathop \sum \limits_{v \in V(G)} |\deg _G v - r|< r(2 + 2l) - 2k$$ , then the edge independence numberβ ₁ (G) is at least (p ? 2l)/2, and (2) ifp is odd, The sharpness of these results is discussed.