TY - JOUR
T1 - Neighbor Sum Distinguishing Total Chromatic Number of Graphs with Lower Average Degree
AU - Huang , Danjun
AU - Bao , Dan
JO - Journal of Mathematical Study
VL - 2
SP - 206
EP - 218
PY - 2023
DA - 2023/06
SN - 56
DO - http://doi.org/10.4208/jms.v56n2.23.06
UR - https://global-sci.org/intro/article_detail/jms/21836.html
KW - Neighbor sum distinguishing total coloring, combinatorial nullstellensatz, maximum average degree.
AB - <p style="text-align: justify;">For a given simple graph $G = (V(G),E(G)),$ a proper total-$k$-coloring $c :
V(G)∪E(G)→ \{1,2,...,k\}$ is neighbor sum distinguishing if $f(u)&nbsp;≠ f(v)$ for each edge $uv ∈ E(G),$ where $f(v) = \sum_{wv∈E(G)} c(wv)+c(v).$ The smallest integer $k$ in such a coloring of $G$ is the neighbor sum distinguishing total chromatic number, denoted by $\chi^{&#39;&#39;}_{\sum} (G).$ It has been conjectured that $\chi ^{&#39;&#39;}_{\sum} (G) ≤ ∆(G)+3$ for any simple graph $G.$ Let $mad (G)=max\{ \frac{2|E(H)|}{ |V(H)|} :H⊆G\}$ be the maximum average degree of $G.$ In this paper, by
using the famous Combinatorial Nullstellensatz, we prove $\chi^{&#39;&#39;} _{\sum}(G) ≤ max\{9,∆(G)+2\}$ for any graph $G$ with $mad (G)&lt;4.$ Furthermore, we characterize the neighbor sum distinguishing total chromatic number for every graph $G$ with $mad (G)&lt;4$ and $∆(G)≥8.$</p>