Abstract
Tashkinov-trees have been used as a tool for proving bounds on the chromatic index, and are becoming a fundamental tool for edge-coloring. Was its publication in a language different from English an obstacle for the accessibility of a clean and complete proof of Tashkinov’s fundamental theorem?
Tashkinov’s original Russian paper offers a clear presentation of this theorem and its proof. The theorem itself has been well understood and successfully applied, but the proof is more difficult. It builds a truly amazing recursive machine, where the various cases necessitate a refined and polished analysis to fit into one another with surprising smoothness and accuracy. The difficulties were brilliantly unknotted by the author, deserving repeated attention.
The present work is the result of reading, translating, reorganizing, rewriting, completing, shortcutting and annotating Tashkinov’s proof. It is essentially the same proof, though with non-negligible communicational differences; for instance, completing it where it appeared to be necessary and simplifying it when it appeared to be possible, while at the same time trying to adapt it to the habits and taste of the international graph theory community.
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Notes
A first draft of this manuscript (dating from December 2017–January 2018) was proof-read only five years later, in order to provide background material to Guantao Chen’s lecture series at the Cargèse Workshop on Combinatorial Optimization https://www.cargese.org/2022/ on Chen, Jing and Zang’s proof of the Goldberg-Seymour conjecture https://arxiv.org/abs/1901.10316, extensively using Tashkinov trees and some new extensions.
Besides Chen, Jing and Zang’s article, several theses and papers explain partly or completely Tashkinov’s theorem, and most of them use it in an original and efficient way to prove relevant new results on edge-coloring. The most complete treatment of Tashkinov-trees we are aware of is Chapter 5 of the book “Vizing’s Theorem and Goldberg’s Conjecture” by Stiebitz’s, Scheide, Toft, Favrholdt (John Wiley and Sons, 2012).
However, for a full and easier understanding of a proof of Tashkinov’s theorem it appeared to be useful to return to the original Russian text, revise it, and present another English version to graph theorists.
References
Kirstead, H.A.: On the chromatic index of multigraphs without large triangles. J. Combin. Theory Ser. B 36(2), 156–160 (1984)
Tashkinov, V.A.: On an algorithm for the edge coloring of multigraphs [in Russian] Diskr. Anal. Issled. Oper. 1, 7(3), 72–85 (2000). Original ref: B. A. Taшкинов, Oб одном алгоритме раскраски ребëр мул тиграфов, Дискретныĭ Анализ и исследование оператсиĭ, Серия 1, Том 7, 3, (2000) 72–85
Vizing, V.G.: On an estimate of the chromatic class of a \(p\)-graph [Russian]. Diskret. Analiz. 3, 25–30 (1964)
Vizing, V.G.: The chromatic class of a multigraph [Russian]. Kibernetika 3, 29–39 (1965)
Acknowledgements
Thanks are due to Penny Haxell for telling me about Tashkinov-trees. I warmly thank Guantao Chen for his kind encouragement to revise and publish my draft notes, initially meant only for domestic use, and for reading then the manuscript with his sharp eyes. I am highly indebted to Ahmad Abdi for numerous relevant suggestions significantly improving the presentation; Ahmad turned from an interested reader to a friendly but strict, excellent reviewer. This work started as a translation of Tashkinov’s original proof [2], which remained its unique source. Our references are thus only [2] and the cited subset of its bibliography.
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Sebő, A. Tashkinov-Trees: an Annotated Proof. Graphs and Combinatorics 40, 4 (2024). https://doi.org/10.1007/s00373-023-02712-1
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DOI: https://doi.org/10.1007/s00373-023-02712-1