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