# Neighbor sum distinguishing total coloring of 2-degenerate graphs

@article{Yao2017NeighborSD,
title={Neighbor sum distinguishing total coloring of 2-degenerate graphs},
author={Jing Jing Yao and Xiaowei Yu and G. Wang and Changqing Xu},
journal={Journal of Combinatorial Optimization},
year={2017},
volume={34},
pages={64-70}
}
• Published 1 July 2017
• Mathematics
• Journal of Combinatorial Optimization
A proper k-total coloring of a graph G is a mapping from $$V(G)\cup E(G)$$V(G)∪E(G) to $$\{1,2,\ldots ,k\}$${1,2,…,k} such that no two adjacent or incident elements in $$V(G)\cup E(G)$$V(G)∪E(G) receive the same color. Let f(v) denote the sum of the colors on the edges incident with v and the color on vertex v. A proper k-total coloring of G is called neighbor sum distinguishing if $$f(u)\ne f(v)$$f(u)≠f(v) for each edge $$uv\in E(G)$$uv∈E(G). Let $$\chi ''_{\Sigma }(G)$$χΣ′′(G) denote the…
12 Citations
Neighbor sum distinguishing total coloring of planar graphs without 4-cycles
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J. Comb. Optim.
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It is proved that the famous Combinatorial Nullstellensatz is used to prove thatchi _{\Sigma }''(G) ∼2, 10$$χΣ′′ (G)≤max{Δ(G)+2,10} for planar graph G without 4-cycles. Neighbor Sum (Set) Distinguishing Total Choosability of d-Degenerate Graphs • Mathematics Graphs Comb. • 2016 Let$$G=(V,E)$$G=(V,E) be a graph with maximum degree$$\varDelta (G)$$Δ(G) and$$\phi :V\cup E\rightarrow \{1,2,\ldots ,k\}$$ϕ:V∪E→{1,2,…,k} be a proper total coloring of the graph G. Let S(v) Neighbor Sum Distinguishing Total Colorings of Corona of Subcubic Graphs • Mathematics • 2020 A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set$$[k]=\{1,2,\ldots ,k\}$$. Let$$\Sigma (u)$$denote the sum of the color on a vertex u and Neighbor sum distinguishing list total coloring of subcubic graphs • Mathematics J. Comb. Optim. • 2018 Some reducible configurations of NSD list total coloring for general graphs are proposed by applying the Combinatorial Nullstellensatz, presenting that chΣt(G)≤Δ(G)+3 for every subcubic graph G. Neighbor product distinguishing total colorings of corona of subcubic graphs. • Mathematics • 2020 A proper [k]-total coloring c of a graph G is a mapping c from V(G)\bigcup E(G) to [k]=\{1,2,\cdots,k\} such that c(x)\neq c(y) for which x, y\in V(G)\bigcup E(G) and x is Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles • Mathematics Discuss. Math. Graph Theory • 2020 By using the discharging method, it is proved that for any planar graph G without 5-cycles, χΣ″(G)≤max{Δ(G)+2, 10} \chi _\Sigma ^{''} ( G ) isle of 2 and 2 for any graph with maximum degree Δ(G). Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2 A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total ## References SHOWING 1-10 OF 26 REFERENCES Neighbor product distinguishing total colorings • Mathematics J. Comb. Optim. • 2017 It is proved that the conjecture that the smallest value k in such a coloring of G, the complete graphs, cycles, trees, bipartite graphs and subcubic graphs, holds for any simple graph with maximum degree Delta (G). Neighbor sum distinguishing total colorings of planar graphs • Mathematics J. Comb. Optim. • 2015 It is proved that the conjecture that any planar graph with maximum degree at least 13 is a simple graph withmaximum degree 3 is correct. Neighbor Sum (Set) Distinguishing Total Choosability of d-Degenerate Graphs • Mathematics Graphs Comb. • 2016 Let$$G=(V,E)$$G=(V,E) be a graph with maximum degree$$\varDelta (G)$$Δ(G) and$$\phi :V\cup E\rightarrow \{1,2,\ldots ,k\}ϕ:V∪E→{1,2,…,k} be a proper total coloring of the graph G. Let S(v)
On the adjacent vertex-distinguishing total chromatic numbers of the graphs with Δ (G) = 3
It is proved that the adjacent vertex-distinguishing total chromatic number of G, \chi_{at}(G)\leq 6, is correct for all connected graphs with maximum degree three.
Neighbor Sum (Set) Distinguishing Total Choosability Via the Combinatorial Nullstellensatz
• Mathematics
Graphs Comb.
• 2017
This paper uses the famous Combinatorial Nullstellensatz to prove that in both problems the challenging conjectures presume that such colorings exist for any graph G if k≥Δ(G)+3, and if G is not a forest andΔ≥4, then the upper bound of the form varDelta (G)+C for some families of graphs is obtained.
Adjacent vertex-distinguishing edge coloring of 2-degenerate graphs
• Mathematics
J. Comb. Optim.
• 2016
It is proved that if G is a 2-degenerate graph without isolated edges, then χavd′ (G) is the smallest integer for which G admits a proper edge k-coloring such that no pair of adjacent vertices are incident with the same set of colors.
On Adjacent Vertex-distinguishing Total Chromatic Number of Generalized Mycielski Graphs
• Mathematics
• 2017
The adjacent vertex-distinguishing total chromatic number of a graph $G$, denoted by $\chi_{at}(G)$, is the smallest $k$ for which $G$ has a proper total $k$-coloring such that any two adjacent
Neighbor sum distinguishing total colorings of K4-minor free graphs
• Mathematics
• 2013
A total [k]-coloring of a graph G is a mapping ϕ: V (G) ∪ E(G) → {1, 2, …, k} such that any two adjacent elements in V (G)∪E(G) receive different colors. Let f(v) denote the sum of the colors of a
Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree
• Mathematics
• 2014
A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h] = {1, 2, …, h}. Let w(u) denote the sum of the color on a vertex u and colors on all the edges