Journal of East China Normal University(Natural Science) ›› 2024, Vol. 2024 ›› Issue (2): 14-22.doi: 10.3969/j.issn.1000-5641.2024.02.002

• Mathematics • Previous Articles     Next Articles

${\rm{E}} $ -total coloring of cycles and paths which are vertex-distinguished by multiple sets

Xiang’en CHEN(), Jing CAO   

  1. 1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
  • Received:2022-04-24 Online:2024-03-25 Published:2024-03-18

Abstract:

An ${\rm{E}} $ -total coloring of a graph $G $ is an assignment of several colors to all vertices and edges of $G $ such that no two adjacent vertices receive the same color and no edge receive the same color as one of its endpoints. If $f $ is an ${\rm{E}} $ -total coloring of a graph $G $, the multiple color set of a vertex $x $ of $G $ under $f $ is the multiple set composed of colors of $x $ and the edges incident with $x $. If any two distinct vertices of $G $ have distinct multiple color sets under an ${\rm{E}} $ -total coloring $f $ of a graph $G $, then $f $ is called an ${\rm{E}} $ -total coloring of $G $ vertex-distinguished by multiple sets. An ${\rm{E}} $ -total chromatic number of $G $ vertex-distinguished by multiple sets is the minimum number of the colors required in an ${\rm{E}} $ -total coloring of $G $ vertex-distinguished by multiple sets. The ${\rm{E}} $ -total colorings of cycles and paths vertex-distinguished by multiple sets are discussed by use of the method of contradiction and the construction of concrete coloring. The optimal${\rm{E}} $ -total colorings of cycles and paths vertex-distinguished by multiple sets are given and the ${\rm{E}} $ -total chromatic numbers of cycles and paths vertex-distinguished by multiple sets are determined in this paper.

Key words: cycle, path, multiple color set, ${\rm{E}} $ -total coloring, ${\rm{E}} $ -total coloring vertex-distinguished by multiple sets

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