Abstract:

In 1968, Vizing conjectured for any edge chromatic critical graph $G$ with maximum degree $\Delta$ and independence number $\alpha(G)$, $\alpha(G)\leqslant\dfrac{n}{2}$. This conjecture is still open. In this paper, we proved that $\alpha(G)\leqslant\dfrac{7\Delta-6}{12\Delta-6}|V|$ for $\Delta\in\{3, 4, 5, 6\} $ and $\alpha(G)\leqslant \dfrac{4\Delta-3}{7\Delta-3}|V|$ for $\Delta\in\{7, 8, 9\}$.

Key words: edge coloring, critical graphs, independence number