A simple graph G is called \(\varDelta\)-critical if \(\chi '(G) =\varDelta (G) +1\) and \(\chi '(H) \le \varDelta (G)\) for every proper subgraph H of G, where \(\varDelta (G)\) and \(\chi '(G)\) are the maximum degree and the chromatic index of G, respectively. Vizing in 1965 conjectured that any \(\varDelta\)-critical graph contains a 2-factor, which is commonly referred to as Vizing’s 2-factor conjecture; In 1968, he proposed a weaker conjecture that the independence number of any \(\varDelta\)-critical graph with order n is at most n/2, which is commonly referred to as Vizing’s independence number conjecture. Based on a construction of \(\varDelta\)-critical graphs which is called Meredith extension first given by Meredith, we show that if \(\alpha (G')\le (\frac{1}{2}+f(\varDelta ))|V(G')|\) for every \(\varDelta\)-critical graph \(G'\) with \(\delta (G')=\varDelta -1,\) then \(\alpha (G)<\big (\frac{1}{2}+f(\varDelta )(2\varDelta -5)\big )|V(G)|\) for every \(\varDelta\)-critical graph G with maximum degree \(\varDelta ,\) where f is a nonnegative function of \(\varDelta .\) We also prove that any \(\varDelta\)-critical graph contains a 2-factor if and only if its Meredith extension contains a 2-factor.

一个简单的图形ģ称为\（\ varDelta \） -临界如果\（\志'（G）= \ varDelta（G）1 \）和\（\志'（H）\文件\ varDelta（G）\ ）为每个子图正确ħ的ģ，其中\（\ varDelta（G）\）和\（\志“（G）\）是最大程度和的色指数ģ，分别。1965年Vizing推测任何\（\ varDelta \）-临界图都包含2因子，通常称为Vizing的2因子猜想；1968年，他提出了一个更弱的猜想，即任何\（\ varDelta \）的独立性数n阶的临界图最多为n / 2，通常称为Vizing独立数猜想。基于Meredith首先给出的\（\ varDelta \）临界图的构造（称为Meredith扩展），我们证明了如果\（\ alpha（G'）\ le（\ frac {1} {2} + f（ \ varDelta））| V（G'）| \）的每个\（\ varDelta \）临界图\（G'\）带有\（\ delta（G'）= \ varDelta -1，\），然后\（每个\（\ varDelta \）的\ alpha（G）<\ big（\ frac {1} {2} + f（\ varDelta）（2 \ varDelta -5）\ big）| V（G）| \） -最大程度的临界图G\（\ varDelta，\）其中f是\（\ varDelta。\）的非负函数。我们还证明，只要且仅当它的Meredith扩展包含2时，任何\（\ varDelta \）临界图才包含2因子。 -因子。