Counting Packings of List-colorings of Graphs
Hemanshu Kaul, Jeffrey A. Mudrock
TL;DR
The paper introduces and studies counting packings of list colorings, defining $P_ ext{l}^ op(G,q,k)$ and $P^ op(G,q,k)$ and connecting them to the chromatic polynomial and list color function. It proves a general equality $P_ ext{l}^ op(G,q,k)=P^ op(G,q,k)$ for all graphs when $q \\ge nk(k-1)/2 + mk - 1$, extends the result to the special case $k=q$ for trees, and develops a Cartesian-product framework yielding $P^ op(G,q,k)=P(G \\square K_k,q)/k!$. The work also deploys a polynomial-method approach to obtain exponential lower bounds on $P_ ext{l}^ op(G,q)$ for sparse graphs, with concrete corollaries for planar graphs of girth at least $8$, underscoring the enumerative richness of list-packings beyond classical colorings.
Abstract
Given a list assignment for a graph, list packing asks for the existence of multiple pairwise disjoint list colorings of the graph. Several papers have recently appeared that study the existence of such a packing of list colorings. Formally, a proper $L$-packing of size $k$ of a graph $G$ is a set of $k$ pairwise disjoint proper $L$-colorings of $G$ where $L$ is a list assignment of colors to the vertices of $G$. In this note, we initiate the study of counting such packings of list colorings of a graph. We define $P_\ell^\star(G,q,k)$ as the guaranteed number of proper $L$-packings of size $k$ of $G$ over all list assignments $L$ that assign $q$ colors to each vertex of $G$, and we let $P^\star(G,q,k)$ be its classical coloring counterpart. We let $P_\ell^\star(G,q)= P_\ell^\star(G,q,q)$ so that $P_\ell^\star(G,q)$ is the enumerative function for the previously studied list packing number $χ_\ell^\star(G)$. Note that the chromatic polynomial of $G$, $P(G,q)$, is $P^\star(G,q,1)$, and the list color function of $G$, $P_\ell(G,q)$, is $P_\ell^\star(G,q,1)$. Inspired by the well-known behavior of the list color function and the chromatic polynomial, we make progress towards the question of whether $P_{\ell}^\star(G,q,k) = P^\star(G,q,k)$ when $q$ is large enough. Our result generalizes the recent theorem of Dong and Zhang (2023), which improved results going back to Donner (1992), about when the list color function equals the chromatic polynomial. Further, we use a polynomial method to generalize bounds on the list packing number, $χ_\ell^\star(G)$, of sparse graphs to exponential lower bounds (in the number of vertices of $G$) on the corresponding list packing functions, $P_\ell^\star(G,q)$.