Neighbor sum distinguishing total choosability of planar graphs

Qu, Cunquan; Wang, Guanghui; Yan, Guangwu; Yu, Xiaowei

doi:10.1007/s10878-015-9911-9

Cited by 26 publications

(3 citation statements)

References 20 publications

(12 reference statements)

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“…Obviously, Conjecture 2 implied Conjecture 1. Qu et al [10] proved that this conjecture holds for any planar graph G with maximum degree ∆(G) ≥ 13. Wang et al [11] confirmed Conjecture 2 for every planar graph G with maximum degree ∆(G) ≥ 8 but without theta graphs Θ 2,1,2 .…”

Section: Conjecture 1 ([2]mentioning

confidence: 97%

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

Zhang

2021

Mathematics

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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 coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total (Δ+3)-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ≥9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

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Section: Conjecture 1 ([2]mentioning

confidence: 97%

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

Zhang

2021

Mathematics

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“…A conjecture about it is put forward: the neighbor sum distinguishing the total coloration of any simple graph does not exceed its maximum degree plus 3. There are more kinds of literature related to neighbor sum distinguishing total coloring [4][5][6].…”

Section: For a Proper ]mentioning

confidence: 99%

Neighbor Sum Distinguishing Edge (Total) Coloring of Generalized Corona Product

Zhong

Tian

2022

J. Phys.: Conf. Ser.

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The coloring theory of graphs is an important part of graph theory research. The key problem of the coloring theory of graphs is to determine the coloring number of each kind of coloring. Traditional coloring concepts mainly include proper vertex coloring, proper edge coloring, proper total coloring, and so on. In recent years, scholars at home and abroad have put forward some new coloring concepts, such as neighbor vertex distinguishing edge (total) coloring, and neighbor sum distinguishing edge (total) coloring, based on traditional coloring concepts and by adding other constraints. Some valuable results have been obtained, which further enrich the theory of graph coloring. For a proper [k]-edge coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing [k]-edge coloring of G. For a proper [k]-total coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing [k]-total coloring of G . In this paper, the coloring method and coloring index are determined by the process of induction and deduction and the construction of the dyeing method, and then the rationality of the method is verified by inverse proof and mathematical induction. If G is a simple graph with the order n ≥ 5 , and hn = (Hi ) i∈{1,2,…,n} is a sequence of disjoint simple graphs, where every Hi is a simple graph with the order m ≥ 7 . In this paper, we study the neighbor sum distinguishing edge(total) coloring of the generalized corona product G○hn of G and hn . The results are as follows: (1) If G is a path with order n , hn = (Hi ) i∈{1,2,…,n} is an alternating sequence of path and cycle with order m . If n is odd, we have χ Σ ′ ( G ∘ h n ) = m + 3 (2) If G is a path with order n , hn = (Hi ) i∈{1,2,…,n} is an alternating sequence of path and cycle with order m . If n is odd, we have χ Σ ′ ′ ( G ∘ h n ) = m + 4 Due to the late development of neighbor sum distinguishing edge (total) coloring of graphs, the related research results are relatively few. By studying the operation graph of a basic simple graph, we can provide the research basis and reference idea for the corresponding coloring of the general graph class. Therefore, it is of theoretical value to study the neighbor sum distinguishing edge (total) coloring problem of generalized corona products of graphs.

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“…Qu et al [15] proved that Ch ∑ ( ) ≤ Δ( ) + 3 for any planar graph with Δ( ) ≥ 13. Yao et al [16] studied Ch ∑ ( ) of -degenerate graphs.…”

Section: Introductionmentioning

confidence: 99%

Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers

Jumnongnit

Nakprasit

2017

International Journal of Mathematics and Mathematical Sciences

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Let be a graph and : ( ) ∪ ( ) → {1, 2, 3, . . . , } be a -total coloring. Let (V) denote the sum of color on a vertex V and colors assigned to edges incident to V. If ( ) ̸ = (V) whenever V ∈ ( ), then is called a neighbor sum distinguishing total coloring. The smallest integer such that has a neighbor sum distinguishing -total coloring is denoted by tndi ∑ ( ). In 2014, Dong and Wang obtained the results about tndi ∑ ( ) depending on the value of maximum average degree. A -assignment of is a list assignment of integers to vertices and edges with | (V)| = for each vertex V and | ( )| = for each edge . A total--coloring is a total coloring of such that (V) ∈ (V) whenever V ∈ ( ) and ( ) ∈ ( ) whenever ∈ ( ). We state that has a neighbor sum distinguishing total--coloring if has a total--coloring such that ( ) ̸ = (V) for all V ∈ ( ). The smallest integer such that has a neighbor sum distinguishing total--coloring for every -assignment is denoted by Ch ∑ ( ). In this paper, we strengthen results by Dong and Wang by giving analogous results for Ch ∑ ( ).

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Neighbor sum distinguishing total choosability of planar graphs

Abstract: A total-k-coloring of a graph G is a mapping c

Cited by 26 publications

References 20 publications

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

Neighbor Sum Distinguishing Edge (Total) Coloring of Generalized Corona Product

Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers

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