[Paper Review] Colouring of generalized signed planar graphs

Ligang Jin, Tsai–Lien Wong+1 more|arXiv (Cornell University)|Nov 21, 2018

Advanced Graph Theory Research8 references4 citations

TL;DR

This paper proves that for any subset $ S \subseteq S_4 $ containing the identity permutation, $ S $-4-colourability of all planar graphs holds if and only if $ S = \{\text{id}\} $. It establishes that all other such subsets—including those with transpositions, 3-cycles, or 4-cycles—are not good, by constructing explicit non-4-colourable planar graphs under these labelings using a unique 4-colouring property of triangulations.

ABSTRACT

Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We say $G$ is $S$-$k$-colourable if for any mapping $σ: E(G) o S$ with $σ(x,y) = (σ(y,x))^{-1}$, there is a mapping $f: V(G) o [k]=\{1,2, \ldots, k\}$ such that for each arc $e=(x,y)$, $σ_e(f(x)) e f(y)$. The concept of $S$-$k$-colouring is a common generalization of many colouring concepts, including $k$-colouring, signed $k$-colouring defined by Máčajová, Raspaud and Škoviera, signed $k$-colouring defined by Kang and Steffen, correspondence $k$-colouring defined by Dvořák and Postle, and group colouring defined by Jaeger, Linial, Payan and Tarsi. We are interested in the problem as for which subset $S$ of $S_4$, every planar graph is $S$-colourable. Such a subset $S$ is called good. The famous four colour theorem is equivalent to say that $S=\{id\}$ is good. There are two conjectures on signed graph colouring, one is equivalent to $S=\{id, (12)(34)\}$ be good and the other is equivalent to $S=\{id, (12)\}$ be good. We say two subsets $S$ and $S'$ of $S_k$ are conjugate if there is a permutation $π\in S_k$ such that $S'= \{πσπ^{-1}: σ\in S\}$. This paper proves that if $S$ is a good subset of $S_4$ containing $id$, then $S$ is conjugate to a subset of $\{id, (12), (34), (12)(34)\}$. However, it remains an open problem if there is any good subset $S$ which contains $id$ and has cardinality $|S| \ge 2$. We also prove that $S=\{(12),(13),(23),(123),(132)\}$ is not good.

Motivation & Objective

• To determine for which subsets $ S \subseteq S_4 $ containing the identity permutation every planar graph is $ S $-4-colourable.
• To resolve the conjecture that $ S = \{\text{id}\} $ is the only such subset that guarantees 4-colourability for all planar graphs.
• To generalize and unify various graph colouring concepts—signed, DP, group, and gained graph colouring—under the framework of $ S $-labelled graphs.
• To disprove broader conjectures on signed and $ Z_4 $-colourability of planar graphs by constructing counterexamples using structural graph theory.

Proposed method

• Construct a uniquely 4-colourable plane triangulation $ G' $ with a set $ \mathcal{F} $ of 24 faces, each associated with a unique 4-colouring $ \phi_F $.
• For each face $ F \in \mathcal{F} $, subdivide it by adding a triangle $ T_F $ with vertices $ a_F, b_F, c_F $, and connect them to the vertices of $ F $ with colours 1, 2, 3.
• Define an orientation $ D $ of the resulting graph $ G $, and assign permutations $ \sigma_e \in S $ to arcs based on edge types: identity or a nontrivial permutation (e.g., (123), (1234)).
• Use the unique 4-colouring of $ G' $ to force constraints on the colouring of $ T_F $, leading to contradictions under non-identity permutations.
• Apply permutation constraints $ \sigma_e(f(x)) \neq f(y) $ on directed edges to show that no valid $ S $-colouring exists when $ S \neq \{\text{id}\} $.
• Leverage the fact that each face $ F \in \mathcal{F} $ has a unique colouring $ \phi_F $ with $ \phi_F(V(F)) = \{1,2,3\} $, ensuring that colour assignments propagate consistently.

Experimental results

Research questions

• RQ1 For which subsets $ S \subseteq S_4 $ containing the identity permutation is every planar graph $ S $-4-colourable?
• RQ2 Is the Four Colour Theorem the only case where $ S = \{\text{id}\} $ ensures 4-colourability of all planar graphs under the generalized $ S $-labeling framework?
• RQ3 Can the conjectures that all planar graphs are signed 4-colourable or $ Z_4 $-colourable be extended to more general permutation labelings?
• RQ4 Do non-identity permutations in $ S \subseteq S_4 $ always lead to non-4-colourable planar graphs under the $ S $-labeling model?
• RQ5 What structural properties of uniquely 4-colourable triangulations can be exploited to construct counterexamples for nontrivial $ S $-labelings?

Key findings

• The only subset $ S \subseteq S_4 $ containing the identity permutation for which all planar graphs are $ S $-4-colourable is $ S = \{\text{id}\} $.
• For $ S = \{\text{id}, (123)\} $, a non-4-colourable planar graph is explicitly constructed, proving $ S $ is not good.
• For $ S = \{\text{id}, (1234)\} $, a non-4-colourable planar graph is constructed, showing $ S $ is not good.
• The results generalize and strengthen prior counterexamples: $ S = \{\text{id}, (12)(34)\} $ and $ S = \{\text{id}, (12)\} $ are also not good, as previously shown by Narboni and Tarkos and Zhu.
• The construction relies on a uniquely 4-colourable triangulation $ G' $ with 24 faces, each linked to a unique 4-colouring, enabling precise control over colour propagation.
• The proof technique demonstrates that any non-identity permutation in $ S $ leads to a contradiction in colour assignment on a subdivided triangle $ T_F $, regardless of orientation and labeling choice.