## Abstract

Let (Formula presented.) be a multigraph with maximum degree (Formula presented.), chromatic index (Formula presented.), and total chromatic number (Formula presented.). The total coloring conjecture proposed by Behzad and Vizing, independently, states that (Formula presented.) for a multigraph (Formula presented.), where (Formula presented.) is the multiplicity of (Formula presented.). Moreover, Goldberg conjectured that (Formula presented.) if (Formula presented.) and noticed the conjecture holds when (Formula presented.) is an edge-chromatic critical graph. By assuming the Goldberg–Seymour conjecture, we show that (Formula presented.) if (Formula presented.) in this note. Consequently, (Formula presented.) if (Formula presented.) and (Formula presented.) has a spanning edge-chromatic critical subgraph.

Original language | English (US) |
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Pages (from-to) | 182-188 |

Number of pages | 7 |

Journal | Journal of Graph Theory |

Volume | 100 |

Issue number | 1 |

DOIs | |

State | Published - May 2022 |

## Keywords

- chromatic index
- chromatic number
- total chromatic number

## ASJC Scopus subject areas

- Geometry and Topology
- Discrete Mathematics and Combinatorics