Graph Theory
The Edge Coloring Problem in Graphs
DOI:
https://doi.org/10.20873/uft.2675-3588.2026.v7n2.p29-40Keywords:
Graph Coloring, Four Color Problem, Vizing Theorem, Combinatorial Optimization, ModelingAbstract
Graph theory, historically propelled by Francis Guthrie's 1852 conjecture and the eventual proof of the Four Color Theorem, has evolved from a collection of topological curiosities into a set of essential modeling tools. This work specifically targets the Edge Coloring Problem, addressing it through a lens that is both historical and rigorously formal. Initially, the text contextualizes the conceptual shift from map coloring to edge coloring, emphasizing its practical applicability in critical areas such as network optimization and scheduling. The core discussion deepens into an analysis of Vizing's Theorem, which establishes precise boundaries for the chromatic index of simple graphs, positioning it strictly between the maximum degree and the maximum degree plus one. Key lemmas and structural conditions determining whether a graph falls into Class 1 or Class 2 are dissected. By exploring the inherent complexity of this classification, this article serves as a pedagogical reference, clarifying how local adjacency constraints dictate global behavior in complex systems.
Downloads
Published
How to Cite
License
Copyright (c) 2026 Lean de Albuquerque Pereira, Tiago Baborsa de Castro Souza, Daniel Martins da Silva, Tanilson Dias dos Santos
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Authors who publish in this journal agree to the following terms:
- Authors retain copyright and grant the journal the right of first publication, with work simultaneously licensed under the Creative Commons Attribution License (CC BY-NC 4.0), allowing work sharing with acknowledgment of the work's authorship and initial publication in this journal. ;
- Authors are authorized to enter additional contracts separately for the non-exclusive distribution of the version of the work published in this journal (eg, publishing in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal;
- Authors are allowed and encouraged to post and distribute their work online (eg, in institutional repositories or on their personal page) at any point after the editorial process;
- In addition, the AUTHOR is informed and agrees with the journal that, therefore, his paper may be incorporated by the AJCEAM into existing or existing scientific information systems and databases (indexers and databases). in the future (indexers and future databases), under the conditions defined by the latter at all times, which will involve at least the possibility that the holders of these databases may perform the following actions on the paper:
- Reproduce, transmit and distribute the paper in whole or in part in any form or means of existing or future electronic transmission, including electronic transmission for research, viewing and printing purposes;
- Reproduce and distribute all or part of the article in print;
- Translate certain parts of the paper;
- Extract figures, tables, illustrations, and other graphic objects and capture metadata, captions, and related article for research, visualization, and printing purposes;
- Transmission, distribution, and reproduction by agents or authorized by the owners of database distributors;
- The preparation of bibliographic citations, summaries and indexes and related capture references from selected parts of the paper;
- Scan and/or store electronic article images and text.
