Improper coloring of toroidal graphs

2025-09-19

AI Summary

This paper investigates defective coloring of toroidal graphs—partitioning vertices into color classes such that each induced monochromatic subgraph has bounded maximum degree (the defect). Employing Euler characteristic analysis, genus-based arguments, constructive proofs, and the minimal counterexample method, the authors establish several novel results: (i) all toroidal graphs are $(0,0,0,4)$-colorable, significantly improving the prior upper bound of $(0,0,0,14)$; (ii) all $6$-regular toroidal graphs—except $K_7$ and $T_{11}$—are $(0,0,0,1)$-colorable; (iii) graphs embeddable on the projective plane $N_1$ are $(0,0,0,2)$-colorable; and (iv) introducing *-defective coloring (where one vertex may violate the defect constraint), they prove $(0,0,0,0,0,1^*)$- and $(0,0,0,1^*,1^*)$-colorability. These contributions advance the structural partitioning theory of toroidal graphs and defective coloring frameworks.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesMachine Learning: Graph-based Machine LearningKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsResponsible Web: Measurement, analysis, and circumvention of Web censorshipWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A graph $G$ is called $(d_1,dots,d_k)$-colorable if its vertices can be partitioned into $k$ sets $V_1,dots,V_k$ such that $Δ(langle V_i angle_G)leq d_i, iin {1,dots, k}$. If $d_1 = dots = d_k = m$ we say that $G$ is $k$-colorable with defect $m$. A coloring with at least one $d_i, iin {1,dots, k}$, greater than $0$ is called an improper coloring. It is known that toroidal graphs are properly $7$-colorable, therefore they are $7$-colorable with defect $0$. It was also proved that toroidal graphs are $5$-colorable with defect $1$ and $3$-colorable with defect $2$. The question whether they are $4$-colorable with defect $1$ remains open. In this paper we focus on improper coloring of toroidal graphs with values of defects being not all equal. We prove that these graphs are $(0,0,0,0,0,1^*)$-colorable, $(0,0,0,0,2)$-colorable and $(0,0,0,1^*,1^*)$-colorable (a star means that there is an improper coloring in which subgraph induced by the corresponding color class contains at most one edge). Choi and Esperet in [Improper coloring of graphs on surfaces, J. Graph Theory $91(1),(2019), 16-34$] proved that every graph of Euler genus $eg > 0$ is $(0, 0, 0, 9eg - 4)$-colorable. From this result it follows that toroidal graphs are $(0,0,0,14)$-colorable. We decreased the value $14$ and proved that toroidal graphs are $(0,0,0,4)$-colorable. We also show that all 6-regular toroidal graphs except $K_7$ and $T_{11}$ are $(0,0,0,1)$-colorable. Finally, we discuss the colorability of graphs embeddable on $N_1$ and show that they are $(0,0,0,2)$-colorable.

Problem

• Investigating improper coloring of toroidal graphs with unequal defects
• Reducing required defect values for (0,0,0,d)-colorability in toroidal graphs
• Determining colorability conditions for 6-regular toroidal graphs and N_1-embeddable graphs

Innovation

• Toroidal graphs are (0,0,0,4)-colorable
• 6-regular toroidal graphs are (0,0,0,1)-colorable
• Graphs embeddable on N1 are (0,0,0,2)-colorable