Maximizing Satisfied Vertex Requests in List Coloring

TL;DR

This paper investigates the concept of list flexibility in graph coloring, establishing bounds for complete multipartite graphs and exploring epsilon flexibility in bipartite graphs, connecting to existing list coloring theories.

Contribution

It proves that the list flexibility number of complete multipartite graphs does not exceed their coloring number and characterizes list epsilon flexibility for small bipartite graphs, linking to prior asymmetric list coloring research.

Findings

List flexibility number of complete multipartite graphs is at most their coloring number.

Complete characterization of list epsilon flexibility for K_{m,n} with m in {1,2}.

Connections made to asymmetric list coloring in bipartite graphs.

Abstract

Suppose $G$ is a graph and $L$ is a list assignment for $G$. A request of $L$ is a function $r$ with nonempty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$. The triple $(G,L,r)$ is $\epsilon$-satisfiable if there exists a proper $L$-coloring $f$ of $G$ such that $f(v) = r(v)$ for at least $\epsilon|D|$ vertices in $D$. We say $G$ is $(k, \epsilon)$-flexible if $(G,L',r')$ is $\epsilon$-satisfiable whenever $L'$ is a $k$-assignment for $G$ and $r'$ is a request of $L'$. It is known that a graph $G$ is not $(k, \epsilon)$-flexible for any $k$ if and only if $\epsilon > 1/ \rho(G)$ where $\rho(G)$ is the Hall ratio of $G$. The list flexibility number of a graph $G$, denoted $\chi_{\ell flex}(G)$, is the smallest $k$ such that $G$ is $(k,1/ \rho(G))$-flexible. A fundamental open question on list flexibility numbers asks: Is there a graph with list flexibility number greater than its coloring number? In this paper, we show that the list flexibility number of any complete multipartite graph $G$ is at most the coloring number of $G$. We also initiate the study of list epsilon flexibility functions of complete bipartite graphs which was first suggested by Kaul, Mathew, Mudrock, and Pelsmajer in 2024. Specifically, we completely determine the list epsilon flexibility function of $K_{m,n}$ when $m \in \{1,2\}$ and establish some additional bounds for small $m$. Our proofs reveal a connection to list coloring complete bipartite graphs with asymmetric list sizes which is a topic that was explored by Alon, Cambie, and Kang in 2021.