The Total Coloring and the Adjacent Vertex-distinguishing Total Coloring of Generalized Mycielski's Graph and Similar Generalized Mycielski's Graph
Lei Sun · Advances in Mathematics · 2010
Mycielski's graph was first introduced by Mycielski in 1955,generalized Mycielski's graph was introduced the natural generalization of Mycielski's graph by Peter Cher Dor Lam,Lin Wensong etc.in 2003,and studied circular chromatic number of it.At present, there were some results of basic properties of generalized Mycielski's graph and vertex chromatic number,fractional chromatic number,circular chromatic number and so on.In this paper,we define similar generalized Mycielski's graph which is another generalization of generalized Mycielski's graph,and we deal with the relations between the basic graphs and their generalized Mycielski's graph or similar generalized Mycielski's graph on the total coloring and the adjacent vertex-distinguish total coloring.And we obtain some sufficient conditions with which the constructional graphs satisfy the total coloring conjectrue and adjacent vertex-distinguish total coloring conjectrue,and with which they reach lower bound of the total chromatic number and the adjacent vertex-distinguishing total chromatic number.