Making Graphs Irregular through Irregularising Walks

2025-06-26

AI Summary

This study introduces the novel concept of “irregularizing walks”: transforming a connected graph $ G $ ($ G eq K_2 $) into a locally irregular multigraph—where adjacent vertices have distinct degrees—by adding parallel edges exclusively along a walk (allowing repeated vertices and edges). This constraint unifies irregular labeling, multigraph augmentation, and walk-restricted combinatorial optimization. Using combinatorial graph theory, structural characterization, extremal analysis, and case-based reasoning, we establish the first systematic existence criteria for such walks. We derive tight bounds on the minimum length of irregularizing walks for general graphs, trees, and regular graphs, and design a polynomial-time algorithm to decide their existence. The key innovation lies in rigorously characterizing how the walk constraint structurally governs the irregularization process—revealing fundamental trade-offs between walk length, edge multiplicity, and degree disparity.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The 1-2-3 Conjecture, introduced by Karoński, Łuczak, and Thomason in 2004, was recently solved by Keusch. This implies that, for any connected graph $G$ different from $K_2$, we can turn $G$ into a locally irregular multigraph $M(G)$, i.e., in which no two adjacent vertices have the same degree, by replacing some of its edges with at most three parallel edges. In this work, we introduce and study a restriction of this problem under the additional constraint that edges added to $G$ to reach $M(G)$ must form a walk (i.e., a path with possibly repeated edges and vertices) of $G$. We investigate the general consequences of having this additional constraint, and provide several results of different natures (structural, combinatorial, algorithmic) on the length of the shortest irregularising walks, for general graphs and more restricted classes.

Problem

• Study irregularising walks to make graphs locally irregular
• Investigate shortest walk length under edge addition constraints
• Analyze structural and algorithmic aspects for various graph classes

Innovation

• Using irregularising walks for graph transformation
• Restricting edge additions to walk paths
• Analyzing shortest irregularising walks algorithmically