List Packing and Correspondence Packing of Planar Graphs

TL;DR

The paper investigates disjoint colorings in planar graphs via list packing and correspondence packing, defining $χ^{⋆}_{\ell}(G)$ and $χ^{⋆}_{c}(G)$ as the minimum packing size that works for all k-assignments. It proves tight upper bounds depending on girth: $χ^{⋆}_{c}(G) ≤ 8$ for all planar graphs, $≤5$ for planar graphs with girth at least 4, and $≤4$ for girth at least 5 (optimal in the latter two cases). The approach combines Hall's theorem, auxiliary bigraphs, and a suite of matching lemmas for $(8,3)$- and $(8,4)$-bigraphs with discharging arguments to control structure in counterexamples. The paper also gives outerplanar graphs with $χ^{⋆}_{\ell}(G)=4$ and discusses the correspondence-packing analogue $χ^{⋆}_{c}$, culminating in several open problems about the exact values and asymptotics in various graph classes and embeddings.

Abstract

For a graph $G$ and a list assignment $L$ with $|L(v)|=k$ for all $v$, an $L$-packing consists of $L$-colorings $\varphi_1,\cdots,\varphi_k$ such that $\varphi_i(v)\ne\varphi_j(v)$ for all $v$ and all distinct $i,j\in\{1,\ldots,k\}$. Let $χ^{\star}_{\ell}(G)$ denote the smallest $k$ such that $G$ has an $L$-packing for every $L$ with $|L(v)|=k$ for all $v$. Let $\mathcal{P}_k$ denote the set of all planar graphs with girth at least $k$. We show that (i) $χ^{\star}_{\ell}(G)\le 8$ for all $G\in \mathcal{P}_3$ and (ii) $χ^{\star}_{\ell}(G)\le 5$ for all $G\in \mathcal{P}_4$ and (iii) $χ^{\star}_{\ell}(G)\le 4$ for all $G\in \mathcal{P}_5$. Part (i) makes progress on a problem of Cambie, Cames van Batenburg, Davies, and Kang. We also construct outerplanar graphs $G$ such that $χ^{\star}_{\ell}(G)=4$, which matches the known upper bound $χ^{\star}_{\ell}(G)\le 4$ for all outerplanar graphs. Finally, we consider the analogue of $χ^{\star}_{\ell}$ for correspondence coloring, $χ^{\star}_c$. In fact, all bounds stated above for $χ^{\star}_{\ell}$ also hold for $χ^{\star}_c$.

Figures (15)

• Figure 1: Left: $H_{v_k}$. Center and right: 1-factors $M'_k$ and $M"_k$ in $H_{v_k}$.
• Figure 2: $H$ contains two edge-disjoint 1-factors. (These appear in the 2-factor formed by deleting edge $uw$.)
• Figure 3: Left: The original graph $H_v$. Right: The new $H_v$, after repacking $y$.
• Figure 4: Left: The edges of $H_v$ in bold, together with the edges of $M_{n_1}$ (plain), the edges of $M_{n_2}$ (dashed), and the edges of $M_{n_3}$ (wavy). Right: $H_v+M_y$ contains the matchings $(1,2,3,4)$ (bold), $(3,1,2,4)$ (plain), and $(3,2,4,1)$ (dashed).
• Figure 5: Left: $H_v^1$ decomposes into the 1-factors $(1,4,2,3)$ and $(2,1,3,4)$. Center: $H^2_v$ decomposes into the 1-factors $(2,1,4,3)$ and $(3,4,2,1)$. Right: $H_v^3$ decomposes into the 1-factors $(1,2,4,3)$ and $(2,4,3,1)$.

Theorems & Definitions (46)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4
• Proposition 5
• proof
• Corollary 6
• proof
• Example 7
• proof