Neighbor Sum Distinguishing Total Chromatic Number of Graphs with Lower Average Degree

For a given simple graph G = (V(G),E(G)), a proper total -k-coloring c : V(G) UE(G)-* {1,2,...,k} is neighbor sum distinguishing if f (u) = f (v) for each edge uv E E(G), where f (v) = n-ary sumation wvEE(G)c(wv)+c(v). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by & chi;& PRIME;& sigma;& PRIME;(G). It has been conjectured that & chi;& PRIME;& sigma;& PRIME;(G) < increment (G)+3 for any simple graph G. Let mad(G) =max{ 2|E(H)| |V(H)| : H C G } be the maximum average degree of G. In this paper, by using the famous Combinatorial Nullstellensatz, we prove & chi;& PRIME;& sigma;& PRIME; (G) < max{9, increment (G) +2} for any graph G with mad(G)<4. Furthermore, we characterize the neighbor sum distinguishing total chromatic number for every graph G with mad(G) < 4 and increment (G) > 8.