# On uniquely k-list colorable planar graphs, graphs on surfaces, and regular graphs

## TL;DR

This paper investigates the properties of uniquely k-list colorable graphs, providing bounds for graphs on surfaces, planar graphs, and regular graphs, expanding understanding beyond previously studied multipartite graphs.

## Contribution

It introduces bounds on property M(k) for graphs on surfaces and regular graphs, and initiates a general study on list size variations and graph embeddings.

## Key findings

- Bounds on M(k) for graphs on surfaces
- New results on planar graphs
- Initial exploration of regular graphs and list size variations

## Abstract

A graph $G$ is called uniquely k-list colorable (U$k$LC) if there exists a list of colors on its vertices, say $L=\lbrace S_v \mid v \in V(G) \rbrace $, each of size $k$, such that there is a unique proper list coloring of $G$ from this list of colors. A graph $G$ is said to have property $M(k)$ if it is not uniquely $k$-list colorable. Mahmoodian and Mahdian characterized all graphs with property $M(2)$. For $k\geq 3$ property $M(k)$ has been studied only for multipartite graphs. Here we find bounds on $M(k)$ for graphs embedded on surfaces, and obtain new results on planar graphs. We begin a general study of bounds on $M(k)$ for regular graphs, as well as for graphs with varying list sizes.