On adjacent-vertex-distinguishing total coloring of graphs

On the Total-Neighbor-Distinguishing Index by Sums

“…Zhang, Chen, Li, Yao, Lu and Wang in [11] investigated a proper total coloring of G, but to every vertex v they assigned a set S(v) of colors of the edges incident to v and the color of v. By χ a (G) or by χ at they denoted the smallest number k of colors so that there exists a proper total coloring with k colors that distinguishishes adjacent vertices by sets (i.e., S(u) is different from S(v) for every pair of adjacent vertices u, v). We propose to denote this index by tndi(G)-the total-neighbor-distinguishing index.…”

“…First, let n be even. We know from [11] that tndi(K n ) = n + 1. So there exists a proper total coloring distinguishing adjacent vertices by sets such that for any vertex, exactly one number is missing in its set.…”

On multiset colorings of generalized corona graphs

“…Further, in [4], Chartrand et al studied the situation when the vertex coloring may not be proper. Please refer to [1], [2], [3], [7], [10], [12] for more related literatures.…”

Graph Colorings Applied in Scale-Free Networks

“…These coloring is presented in the environment about how to solve the practical problems meet in computer science (such as the description of contacting between point and point in space database), up to now, it is concerned by more and more researchers. Vertex-distinguishing total coloring and adjacent vertex distinguishing total coloring are studied in [5]- [7]. Conjecture 1.…”

“…Conjecture 1. [7] Every simple graph G on order n≥2 has its adjacent vertex distinguishing total chromatic number (avdtc chromatic number) χ″ as (G)≤Δ(G)+3.…”