Refined list version of Hadwiger’s Conjecture

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 $\lambda$-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

(4, 2)-Choosability of Planar Graphs with Forbidden Structures

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any ℓ ∈ {3, 4, 5, 6, 7}, a planar graph is 4-choosable if it is ℓ-cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded ℓ-cycle is an ℓ-cycle with one additional edge. We demonstrate for each ℓ ∈ {5, 6, 7} that a planar graph is (4, 2)-choosable if it does not contain chorded ℓ-cycles

Minimum non-chromatic-λ\lambda-choosable graphs

For a multi-set $\lambda=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if $G$ is $L$-colourable for any $\lambda$-list assignment $L$ of $G$. The concept of $\lambda$-choosability puts $k$-colourability and $k$-choosability in the same framework: If $\lambda = \{k\}$, then $\lambda$-choosability is equivalent to $k$-choosability; if $\lambda$ consists of $k$ copies of $1$, then $\lambda$-choosability is equivalent to $k$-colourability. If $G$ is $\lambda$-choosable, then $G$ is $k_{\lambda}$-colourable. On the other hand, there are $k_{\lambda}$-colourable graphs that are not $\lambda$-choosable, provided that $\lambda$ contains an integer larger than $1$. Let $\phi(\lambda)$ be the minimum number of vertices in a $k_{\lambda}$-colourable non-$\lambda$-choosable graph. This paper determines the value of $\phi(\lambda)$ for all $\lambda$.Comment: 13 page

Minimum non-chromatic-λ-choosable graphs

For a multi-set $\lambda=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if $G$ is $L$-colourable for any $\lambda$-list assignment $L$ of $G$. The concept of $\lambda$-choosability puts $k$-colourability and $k$-choosability in the same framework: If $\lambda = \{k\}$, then $\lambda$-choosability is equivalent to $k$-choosability; if $\lambda$ consists of $k$ copies of $1$, then $\lambda$-choosability is equivalent to $k$-colourability. If $G$ is $\lambda$-choosable, then $G$ is $k_{\lambda}$-colourable. On the other hand, there are $k_{\lambda}$-colourable graphs that are not $\lambda$-choosable, provided that $\lambda$ contains an integer larger than $1$. Let $\phi(\lambda)$ be the minimum number of vertices in a $k_{\lambda}$-colourable non-$\lambda$-choosable graph. This paper determines the value of $\phi(\lambda)$ for all $\lambda$