Flexible DP 3-coloring of sparse multigraphs

TL;DR

This work extends the concept of fractional flexibility from list coloring to DP-coloring in sparse multigraphs. By introducing a potential-based framework driven by a partition of vertices into $ ho$-classes and $h_{ ho}$-lists, the authors connect DP-coloring feasibility to fractional $(H,L)$-packings and discharging arguments. They prove that every loopless multigraph with $ ext{mad}(G)<3$ is $rac{1}{5}$-flexibly DP $3$-colorable, with an explicit infinite exceptional family $\mathcal{I}$ and tightness results, and they provide a stronger, potential-driven statement that governs minimal counterexamples. In addition, the paper yields a negative answer to a question of Dvořák–Norin–Postle about flexibility for list coloring in the DP-coloring setting, highlighting nuanced limitations of flexible colorings in this generalized framework. The combination of potential methods, fractional packings, and discharging offers a robust toolkit for analyzing DP-coloring flexibility in sparse multigraphs and informs the boundary between flexible colorability and structural obstructions.

Abstract

A \emph{request} on a graph assigns a preferred color to a subset of the vertices. A graph $G$ is \emph{$ε$-flexibly $k$-choosable} if for every $k$-list assignment $L$ and every request $r$ on $G$, there is an $L$-coloring such that an $ε$-fraction of the requests are satisfied. This notion was introduced in 2019 by Dvořák, Norin, and Postle, who also proved important properties of flexible colorings and posed several natural problems. However, the weighted version of this problem is a special case of the much older problem of fractional hypergraph matchings, introduced by Lovász in 1975. We study flexibly DP-colorable multigraphs. We prove that every loopless multigraph with maximum average degree less than $3$ is $\frac{1}{5}$-flexibly DP $3$-colorable, except for an infinite family of multigraphs that we completely characterize. The constant $ε= \frac 15$ is best possible in the weighted setting, as shown by an infinite family of tight examples. Our result follows from a stronger statement in terms of potential. We also provide a family of graphs that gives a negative answer to a question by Dvořák, Norin, and Postle regarding flexibility for list coloring in the setting of DP-coloring.

Figures (5)

• Figure 1: The multigraphs $I_1$, $I_2$, and $I_3$ are shown from left to right. Each multigraph $I_{m+1}$ is obtained from $I_m$ by performing two subdivisions on any edge of multiplicity $1$ and then adding a second edge joining the two new vertices. Each of them has maximum average degree less than $3$ and is not flexibly DP $3$-colorable.
• Figure 2: The graphs $J_1$, $J_2$, and $J_3$ are shown from left to right. Each graph $J_{m+1}$ is obtained from $J_m$ by performing two subdivisions on an edge of multiplicity $1$ not incident with $v_{2m+1}$ and then adding a new edge joining the two new vertices.
• Figure 3: A $2$-degenerate graph that is not flexibly DP $3$-colorable.
• Figure 5: On the left is the house graph $H_5$, and on the right is its subgraph $H_4$. If $u$ and $v$ are deleted from $H_5$ and an edge is added from $u'$ to $v'$, as in \ref{['lem:kck']}, then $I_3$ is obtained.
• Figure 6: A cover $(H',L')$ of the graph $G'$ in the proof of \ref{['lem:I5']}, along with the bipartite graph $B$ constructed from $(H',L')$. For each $x \in V(G')$, $L'(x)$ is shown with $1_x$ on top and $3_x$ on the bottom. Here, as $1_{u^*}$ is the unique vertex of $L'(u^*)$ that does not belong to a $4$-cycle in $E_{H'}(L'(u^*), L'(w))$, $j_{u^*} = 1_{u^*}$. Each pair $(i_u, i'_v) \in K$ is adjacent exactly to those $c \in J$ for which $c$ is adjacent to neither $i_u$ nor $i'_v$ in $H'$.

Theorems & Definitions (89)

• Theorem 1.2: Dvořák, Norin, Postle 2019DvNoPo
• Theorem 1.3: Dvořák, Norin, Postle 2019DvNoPo
• Lemma 1.4: 2019DvNoPo
• proof
• Theorem 1.5
• Theorem 1.6
• Theorem 2.2
• proof
• proof
• proof