Let $\lambda(G)$ be the smallest number of vertices that can removed from a non-empty graph $G$ so resulting has smaller maximum degree. $\lambda_{\rm e}(G)$ edges for same purpose. $k$ degree $G$, let $t$ $k$, $M(G)$ set $n$ in closed neighbourhood $M(G)$, and $m$ incident to $M(G)$. Fenech author showed $\lambda(G) \leq \frac{n+(k-1)t}{2k}$, they essentially $\lambda (G) n \left ( 1- \frac{k}...

an acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. the acyclic chromatic index of a graph $g$ denoted by $chi_a '(g)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. the maximum degree in $g$ denoted by $delta(g)$, is the lower bound for $chi_a '(g)$. $p$-cuts introduced in this paper acts as a powerfu...