Refined list version of Hadwiger's conjecture

Abstract

Assume $\lambda=\{k_1,k_2, \ldots, k_q\}$ is a partition of $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a $k_\lambda$-list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into $|\lambda|= q$ sets $C_1,C_2,\ldots,C_q$ such that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is \emph{$\lambda$-choosable} if $G$ is $L$-colourable for any $\lambda$-list assignment $L$ of $G$. The concept of $\lambda$-choosability is a refinement of choosability that puts $k$-choosability and $k$-colourability in the same framework. If $|\lambda|$ is close to $k_\lambda$, then $\lambda$-choosability is close to $k_\lambda$-colourability; if $|\lambda|$ is close to $1$, then $\lambda$-choosability is close to $k_\lambda$-choosability. This paper studies Hadwiger's Conjecture in the context of $\lambda$-choosability. Hadwiger's Conjecture is equivalent to saying that every $K_t$-minor-free graph is $\{1 \star (t-1)\}$-choosable for any positive integer $t$. We prove that for $t \ge 5$, for any partition $\lambda$ of $t-1$ other than $\{1 \star (t-1)\}$, there is a $K_t$-minor-free graph $G$ that is not $\lambda$-choosable. We then construct several types of $K_t$-minor-free graphs that are not $\lambda$-choosable, where $k_\lambda - (t-1)$ gets larger as $k_\lambda-|\lambda|$ gets larger. In partcular, for any $q$ and any $\epsilon > 0$, there exists $t_0$ such that for any $t \ge t_0$, for any partition $\lambda$ of $\lfloor (2-\epsilon)t \rfloor$ 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 colouring.