Maximizing Satisfied Vertex Requests in List Coloring
Timothy Bennett, Michael C. Bowdoin, Haley Broadus, Daniel Hodgins, Jeffrey A. Mudrock, Adam K. Nusair, Gabriel Sharbel, Joshua Silverman
TL;DR
The paper advances the theory of flexible list coloring by linking Hall ratio to list flexibility in complete multipartite graphs, proving that the list flexibility number χ_ell flex is bounded by the coloring structure of the graph and establishing ε_ell values for large enough list sizes. The authors present an inductive proof showing χ_ell flex(K_{n1,...,nk}) ≤ n1 + ... + n_{k-1} + 1 and show that ε_ell(K_{m,n},t) equals 1/2 in several small/mid regimes, with complete determinations for K_{2,n} and near-complete results for K_{3,n}. They also provide a thorough analysis of (3,2)-choosability for complete bipartite graphs, connecting to asymmetric list-size coloring. These results illuminate when flexibility gains are possible and reveal structural ties between list coloring, degree/chromatic parameters, and asymmetric lists, contributing toward the broader open questions about the interplay between χ_ell flex and col(G).
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 $ε$-satisfiable if there exists a proper $L$-coloring $f$ of $G$ such that $f(v) = r(v)$ for at least $ε|D|$ vertices in $D$. We say $G$ is $(k, ε)$-flexible if $(G,L',r')$ is $ε$-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 $(k, ε)$-flexible for any $k$ if and only if $ε> 1/ ρ(G)$ where $ρ(G)$ is the Hall ratio of $G$. The list flexibility number of a graph $G$, denoted $χ_{\ell flex}(G)$, is the smallest $k$ such that $G$ is $(k,1/ ρ(G))$-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.