A step towards finding the analog of the Four-Color Theorem for (n,m)-graphs

2024-09-09

arXiv.org

AI Summary

This paper addresses the maximum number of vertices in a planar $(n,m)$-complete graph for $(n,m) eq (0,1)$, establishing the exact upper bound $3(2n+m)^2 + (2n+m) + 1$ and proving its tightness. Methodologically, it introduces the first precise upper bound on the clique number of planar $(n,m)$-graphs, combining combinatorial extremal analysis, graph homomorphism theory, structural induction, and constructive proof techniques to fully resolve the conjecture by Bensmail et al. regarding clique number bounds. The key contribution lies in successfully generalizing the clique-number analysis—originally rooted in the Four Color Theorem—to the mixed directed–undirected labeling graph model, thereby laying a foundational cornerstone for mixed-graph coloring theory. Moreover, the work provides a complete characterization of the size limit of planar $(n,m)$-complete graphs, bridging classical planarity constraints with modern hybrid graph structures.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

An extit{$(n,m)$-graph} $G$ is a graph having both arcs and edges, and its arcs (resp., edges) are labeled using one of the $n$ (resp., $m$) different symbols. An extit{$(n,m)$-complete graph} $G$ is an $(n,m)$-graph without loops or multiple edges in its underlying graph such that identifying any pair of vertices results in a loop or parallel adjacencies with distinct labels. We show that a planar $(n,m)$-complete graph cannot have more than $3(2n+m)^2+(2n+m)+1$ vertices, for all $(n,m) eq (0,1)$ and the bound is tight. This answers a naturally fundamental extremal question in the domain of homomorphisms of $(n,m)$-graphs and positively settles a recent conjecture by Bensmail extit{et al.}~[Graphs and Combinatorics 2017]. Essentially, our result finds the clique number for planar $(n,m)$-graphs, which is a difficult problem except when $(n,m)=(0,1)$, answering a sub-question to finding the chromatic number for the family of planar $(n,m)$-graphs.

Problem

• Determine the maximum vertices in planar (n,m)-complete graphs
• Prove tight upper bound for planar (n,m)-clique size
• Confirm Bensmail et al.'s conjecture on graph limits

Innovation

• Planar (n,m)-complete graph vertex bound
• Tight bound for all (n,m) ≠ (0,1)
• Proof of Bensmail et al. conjecture