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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2211.09048 |
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| _version_ | 1866909111626498048 |
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| author | Kaul, Hemanshu Mathew, Rogers Mudrock, Jeffrey A. Pelsmajer, Michael J. |
| author_facet | Kaul, Hemanshu Mathew, Rogers Mudrock, Jeffrey A. Pelsmajer, Michael J. |
| contents | Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose $0 \leq ε\leq 1$, $G$ is a graph, $L$ is a list assignment for $G$, and $r$ is a function with non-empty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$ ($r$ is called a request of $L$). 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 was shown by Dvořák et al. that if $d+1$ is prime, $G$ is a $d$-degenerate graph, and $r$ is a request for $G$ with domain of size $1$, then $(G,L,r)$ is $1$-satisfiable whenever $L$ is a $(d+1)$-assignment. In this paper, we extend this result to all $d$ for bipartite $d$-degenerate graphs.
The literature on flexible list coloring tends to focus on showing that for a fixed graph $G$ and $k \in \mathbb{N}$ there exists an $ε> 0$ such that $G$ is $(k, ε)$-flexible, but it is natural to try to find the largest possible $ε$ for which $G$ is $(k,ε)$-flexible. In this vein, we improve a result of Dvořák et al., by showing $d$-degenerate graphs are $(d+2, 1/2^{d+1})$-flexible. In pursuit of the largest $ε$ for which a graph is $(k,ε)$-flexible, we observe 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 $G$, and we initiate the study of the list flexibility number of a graph $G$, which is the smallest $k$ such that $G$ is $(k,1/ ρ(G))$-flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_09048 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Flexible list colorings: Maximizing the number of requests satisfied Kaul, Hemanshu Mathew, Rogers Mudrock, Jeffrey A. Pelsmajer, Michael J. Combinatorics 05C15 Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose $0 \leq ε\leq 1$, $G$ is a graph, $L$ is a list assignment for $G$, and $r$ is a function with non-empty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$ ($r$ is called a request of $L$). 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 was shown by Dvořák et al. that if $d+1$ is prime, $G$ is a $d$-degenerate graph, and $r$ is a request for $G$ with domain of size $1$, then $(G,L,r)$ is $1$-satisfiable whenever $L$ is a $(d+1)$-assignment. In this paper, we extend this result to all $d$ for bipartite $d$-degenerate graphs. The literature on flexible list coloring tends to focus on showing that for a fixed graph $G$ and $k \in \mathbb{N}$ there exists an $ε> 0$ such that $G$ is $(k, ε)$-flexible, but it is natural to try to find the largest possible $ε$ for which $G$ is $(k,ε)$-flexible. In this vein, we improve a result of Dvořák et al., by showing $d$-degenerate graphs are $(d+2, 1/2^{d+1})$-flexible. In pursuit of the largest $ε$ for which a graph is $(k,ε)$-flexible, we observe 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 $G$, and we initiate the study of the list flexibility number of a graph $G$, which is the smallest $k$ such that $G$ is $(k,1/ ρ(G))$-flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph. |
| title | Flexible list colorings: Maximizing the number of requests satisfied |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2211.09048 |