Let $\lambda=\{k_1,k_2, \ldots, k_q\}$ be a partition of $k_{\lambda} = \sum_{i=1}^q k_i$ . The $\lambda$ list assignment of $G$ is the $k_\lambda$ list assignment $L$ of $G$, and the color set $\bigcup_{v \in V(G)}L(v) $ is $ Set |\lambda|= q$ to $C_1,C_2,\ldots,C_q$ and for each $i$ and each vertex $v$ in $G$, $|L(v) \cap C_i| \gek_i$. If $G$ is $L$-colourable for the $\lambda$ list assignment $L$ of $G$, then $G$ is \emph{$\lambda$-choosable}. say. The concept of $\lambda$-choosability is an refinement of chooseability that puts $k$-choosability and $k$-colourability into the same framework. $\lambda$-choosability is close to $k_\lambda$-colourability if $|\lambda|$ is close to $k_\lambda$. If $|\lambda|$ is close to $1$, $\lambda$-choosability is close to $k_\lambda$-choosability. In this paper, we study his Hadwiger’s conjecture in the context of $\lambda$ selectability. Hadwiger’s conjecture says that every $K_t$-minor-free graph is $\{1 \star (t-1)\}$-choosable for any positive integer $t$ is the same as $K_t$-minor-$ for any partition $\lambda$ of $t-1$ other than $\{1 \star (t-1)\}$ for $t \ge 5$ Free graph $G$ not selectable by \lambda$. Next, we create several types of $K_t$-minor-free graphs that are not $\lambda$ selectable. where $k_\lambda – (t-1)$ grows as $k_\lambda-|\lambda|$ gets. big. In particular, for any $q$ and any $\epsilon > 0$, for any $t \get_0$, for any partition $\lambda$ in $\lfloor (2-\epsilon) and $t_0$ exists. t \rfloor$ with $|\lambda| =q$, there is a $K_t$-minor-free graph that is not $\lambda$-choosable. The $q=1$ case of this result was recently proved by Steiner, and our proof uses a similar argument. We also generalize this result to $(a,b)$-list coloring.
