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• 1 Department of Mathematics Indiana Purdue University 46805 Fort Wayne IN USA
• 2 Department of Mathematics Western Michigan University 49008 Kalamazoo MI USA
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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

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop \sum \limits_{v \in V(G)} |\deg _G v - r|< r(2 + 2l) - 2k$$ \end{document}
, then the edge independence numberβ1(G) is at least (p − 2l)/2, and (2) ifp is odd, The sharpness of these results is discussed.

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