A note on Goldberg's conjecture on total chromatic numbers
A note on Goldberg's conjecture on total chromatic numbers
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关于戈德堡总色数猜想的注解
DOI：
10.1002/jgt.22771
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发表时间：
2021
影响因子：
0.9
通讯作者：
Jing, Guangming
中科院分区：
文献类型：
--
作者：
Cao, Yan;Chen, Guantao;Jing, Guangming
设G是一个具有最大度、色指数和全色数χ″（G）的重图. Behzad和Vizing独立地提出了全染色猜想，证明了对于重图，χ″（G）≤Δ（G）+μ（G）+1，其中是的重数.此外，Goldberg还证明了当G是边色临界图时，如果并注意到猜想成立，则χ″（G）=χ′（G）.在Goldberg-Seymour猜想的假设下，本文证明了：当G为G时，χ″（G）=χ′（G）.因此，若G有一个生成边色临界子图，则χ″（G）=χ′（G）.
Letbe a multigraph with maximum degree, chromatic index, and total chromatic number χ″(G). The total coloring conjecture proposed by Behzad and Vizing, independently, states that χ″(G)≤Δ(G)+μ(G)+1 for a multigraph, whereis the multiplicity of. Moreover, Goldberg conjectured that χ″(G)=χ′(G) ifand noticed the conjecture holds whenis an edge‐chromatic critical graph. By assuming the Goldberg–Seymour conjecture, we show that χ″(G)=χ′(G) ifin this note. Consequently, χ″(G)=χ′(G) ifandhas a spanning edge‐chromatic critical subgraph.