Abstract:Graph coloring with preferences offers a powerful framework for constraint satisfaction problems in which fulfilling every request is impossible but satisfying a guaranteed positive fraction is highly desirable. A \emph{request} on a graph $G$ equipped with a list assignment $L$ assigns to each vertex of some subset $dom(r)\subseteq V(G)$ a preferred color from its list. Following Dvořák, Norin, and Postle (2019), $G$ is \emph{$\varepsilon$-flexibly $k$-choosable} if, for every $k$-list assignment $L$ and every request $r$, there is an $L$-coloring of $G$ that agrees with $r$ on at least $\varepsilon|dom(r)|$ vertices. The corresponding notion for DP-coloring (correspondence coloring) was formalized by Bradshaw, Choi, and Kostochka (2025). Choi, Clemen, Ferrara, Horn, Ma, and Masařík (2022) proved that every planar graph without $4$-cycles and with $3$-cycle distance at least $2$ is $\varepsilon$-flexibly $4$-choosable. We improve the result in two respects: weakening the hypothesis from $3$-cycle distance $\geq 2$ to vertex-disjoint triangles, and strengthening the conclusion from list flexibility to weighted DP-flexibility: \emph{Every simple planar graph without $4$-cycles and without intersecting triangles is weighted $\varepsilon$-flexibly DP-$4$-colorable.} The list size $4$ is sharp: Montassier, Raspaud, and Wang constructed a planar graph without $4$-cycles, $5$-cycles, and intersecting triangles that is not $3$-choosable.
| Comments: | 16 pages. Comments are welcome |
| Subjects: | Combinatorics (math.CO) |
| MSC classes: | 05C15 |
| Cite as: | arXiv:2605.23647 [math.CO] |
| (or arXiv:2605.23647v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.23647 arXiv-issued DOI via DataCite (pending registration) |
From: Gexin Yu [view email]
[v1]
Fri, 22 May 2026 14:00:28 UTC (16 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。