Ewan Davies
In our previous work we defined the list packing number of a graph, which is the least integer k such that there are always k disjoint proper list colourings whenever we have lists all of size k associated to the vertices. We continue our investigation of how the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree $\Delta$ has list packing number at most $\Delta + 1$
The reconfiguration graph $C_k(G)$ for the $k$-colourings of a graph $G$ has a vertex for each proper $k$-colouring of $G$, and two vertices of $C_k(G)$ are adjacent precisely when those $k$-colourings differ on a single vertex of $G$. Much work has focused on bounding the maximum value of the diameter of $(C_k(G))$ over all $n$-vertex graphs $G$. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if $L$ is a list-assignment for a graph $G$ with $|L(
List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a $k$-list-assignment $L$ of a graph $G$, which is the assignment of a list $L(v)$ of $k$ colours to each vertex $v\in V(G)$, we study the existence of $k$ pairwise-disjoint proper colourings of $
Ewan Davies