(4, 2)-Choosability of Planar Graphs with Forbidden Structures

Abstract

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any $\ell \in \{3,4,5,6,7\}$, a planar graph is 4-choosable if it is $\ell$-cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded $\ell$-cycle is an $\ell$-cycle with one additional edge. We demonstrate for each $\ell \in \{5,6,7\}$ that a planar graph is (4, 2)-choosable if it does not contain chorded $\ell$-cycles.

Appendix: Large Reducible Configurations

In the proof of Theorem 8, we demonstrated that no minimal counterexample exists by showing that there exists a reducible configuration $(C,X,{\text {ex}})$ where G contains a copy of C[X] as an induced subgraph (and also the copy agrees with the external degrees). In this appendix, we provide the details that clarify this assumption. By Lemma 1, we can relax the condition that C[X] is an induced subgraph. We will demonstrate that the configurations that appear after some vertices in X are merged (while also preserving the face lengths, vertex degrees, and lack of chorded 5-cycle) result in reducible configurations.

Let $(C,X, {\text {ex}})$ be a reducible configuration and let $\{x_1,x_1'\},\dots ,\{x_t,x_t'\}$ be a list of vertex pairs in X. For these configurations, we may identify some 3-cycles and 5-cycles that are required to be 5-faces (in the context of the proof of Theorem 8). The resulting configuration $(C',X',{\text {ex}})$ where $C'$ and $X'$ are modified from C and X by merging $x_i$ with $x_i'$ and removing any multiedges or loops that result. We say a list $\{x_1,x_1'\},\dots ,\{x_t,x_t'\}$ is valid for $(C,X,{\text {ex}})$ if the resulting configuration $(C',X',{\text {ex}})$ may appear in a planar graph of minimum degree at least four containing no chorded 5-cycle. There are three situations that can occur when we perform this action.

Pairs too close:  If some pair $\{x_i, x_i'\}$ have $d(x_i,x_i') \le 2$, then either we create a loop or a multiedge when merging $x_i$ and $x_i'$. This will reduce the degree of the resulting vertex, in addition to possibly shortening known 3- and 5-cycles. Since distances only decrease as vertices are merged, a pair failing this property will not appear in any valid list of pairs.

Pairs creating chord  If merging $x_i$ and $x_i'$ creates a chorded 5-cycle, then this configuration would not appear in the minimal counterexample from Theorem 8. Since distances only decrease as vertices are merged, a pair failing this property will not appear in any valid list of pairs.

Reducible pairs  If merging $x_i$ and $x_i'$ does not fit in the above two cases, then we will demonstrate that the resulting configuration is reducible. Even if merging one pair of vertices creates a reducible configuration, we need to check all possible lists of pairs that contain that pair.

After considering all pairs that could be identified, observe that in each case there is no set of three or more vertices where every pair can be identified.

In the following tables, we list one of the configurations (C10)–(C21), label the vertices, and list all pairs of vertices into the three categories above. In the case of reducible pairs, we present the contracted graph. Most of these contracted graphs contain a copy of (C1), (C2), (C10), (C11), or (C12). The only exceptions are the contracted graphs derived from (C16), but each of these configurations has an Alon–Tarsi orientation and hence is reducible.