Core Concepts
The Weisfeiler-Leman dimension of a graph is bounded by 0.15n + o(n).
Abstract
This article analyzes the upper bound on the Weisfeiler-Leman dimension of graphs, focusing on structural complexity measurement. The proof involves various techniques to analyze coherent configurations' structure, including recursive proofs and interspace analysis. The Weisfeiler-Leman dimension determines the difficulty of testing isomorphism using combinatorial algorithms.
Introduction
Weisfeiler-Leman (WL) dimension as a measure for graph complexity.
Core Message
WL-dimension of a graph is at most 0.15n + o(n).
Results
Polynomial time isomorphism algorithm for certain graph classes.
Techniques
Reduction in vertices to simplify coherent configurations.
Critical Configurations
Criteria for identifying critical configurations.
Small Fibers and Interspaces
Classification and analysis of small fiber-induced coherent configurations.
Large and Small Fiber Interspaces
Examination of interspaces between large and small fibers.
Inquiry and Critical Thinking Questions
Stats
証明は、WL次元が最大で3/20・n + o(n) = 0.15・n + o(n)であることを示しています。
グラフの頂点数に関する証拠を使用して、WL次元の上限を決定します。
多項式時間同型性アルゴリズムについて述べられています。