Extension of the Gy'arf'as-Sumner conjecture to signed graphs

2025-11-05

AI Summary

研究签色图的平衡染色数问题，通过分析禁止诱导子图的GS集（特别是二元集）结构，证明了当F1为(K3,-)或(K4,-)且F2为特定线性森林时，签色图可进行平衡染色。

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The balanced chromatic number of a signed graph G is the minimum number of balanced sets that cover all vertices of G. Studying structural conditions which imply bounds on the balanced chromatic number of signed graphs is among the most fundamental problems in graph theory. In this work, we initiate the study of coloring hereditary classes of signed graphs. More precisely, we say that a set F = {F_1, F_2, ..., F_l} is a GS (for Gy'arf'as-Sumner) set if there exists a constant c such that signed graphs with no induced subgraph switching equivalent to a member of F admit a balanced c-coloring. The focus of this work is to study GS sets of order 2. We show that if F is a GS set of order 2, then F_1 is either (K_3, -) or (K_4, -), and F_2 is a linear forest. In the case of F_1 = (K_3, -), we show that any choice of a linear forest for F_2 works. In the case of F_1 = (K_4, -), we show that if each connected component of F_2 is a path of length at most 4, then {F_1, F_2} is a GS set.

Problem

• Extend the Gyárfás-Sumner conjecture to signed graphs coloring
• Study GS sets of order two for balanced coloring of signed graphs
• Determine forbidden subgraphs ensuring constant balanced chromatic number

Innovation

• Extends Gyárfás-Sumner conjecture to signed graphs
• Defines GS sets for hereditary signed graph classes
• Characterizes GS sets of order two with forbidden subgraphs