On a conjecture concerning 4-coloring of graphs with one crossing

TL;DR

The authors study a conjecture that every graph with minimum degree at least $5$ and no separating triangles drawn in the plane with at most one crossing is $4$-colorable. They reduce the problem to rainbow precoloring extension on plane graphs with a $4$-cycle outer face and establish equivalences among related conjectures, enabling a unified framework. Using extensive computer enumeration, they verify the conjecture for graphs with at most $28$ vertices and develop a contraction/expansion framework for generating potential counterexamples, finding no rainbow-forbidding candidates and identifying a single diagonal-forbidding diamond, alongside infinitely many bichromatic-forbidding ones. They outline a roadmap to a proof based on adapting Kempe-chain and discharging techniques to this setting, emphasizing the roles of $D$- and $C$-reducible configurations and outlining an Appendix with detailed reducibility arguments.

Abstract

We conjecture that every graph of minimum degree five with no separating triangles and drawn in the plane with one crossing is 4-colorable. In this paper, we use computer enumeration to show that this conjecture holds for all graphs with at most 28 vertices, explore the consequences of this conjecture and provide some insights on how it could be proved.

Figures (6)

• Figure 1: A $5$-critical graph in ${\cal C}$ with a $4$-face $h$.
• Figure 2: $\star$-contractions (1)
• Figure 3: $\star$-contractions (2)
• Figure 4: The constructions of bichromatic-forbidding 4-candidates.
• Figure 5: A troublesome pentagon.

Theorems & Definitions (69)

• Conjecture 1
• Conjecture 2
• Lemma 3
• Corollary 4
• Conjecture 5
• Theorem 6
• Lemma 7
• proof
• Corollary 8
• Lemma 9