Neighbor sum distinguishing total colorings of planar graphs

Li, Hualong; Ding, Laihao; Liu, Bingqiang; Wang, Guanghui

doi:10.1007/s10878-013-9660-6

Cited by 56 publications

(8 citation statements)

References 18 publications

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“…For a given vertex v, the sum c(v) + d c (v) shall be called its total sum and denoted by d t c (v). In [33] it was conjectured that χ ′′ Σ (G) ≤ ∆(G) + 3 for every graph G, see [27,28,33] for partial results concerning this. Note that proving this apparently very challenging conjecture would require improving the following best known upper bound due to Molloy and Reed concerning the well known Total Colouring Conjecture (that χ ′′ (G) ≤ ∆(G) + 2 for every G) posed by Vizing [40] and independently by Behzad [7].…”

Section: Conjecture 1 ([16]mentioning

confidence: 99%

A note on asymptotically optimal neighbour sum distinguishing colourings

Przybyło

2019

European Journal of Combinatorics

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The least k admitting a proper edge colouring c : E → {1, 2, . . . , k} of a graph G = (V, E) without isolated edges such that e∋u c(e) = e∋v c(e) for every uv ∈ E is denoted by χ ′ Σ (G). It has been conjectured that χ ′ Σ (G) ≤ ∆ + 2 for every connected graph of order at least three different from the cycle C 5 , where ∆ is the maximum degree of G. It is known that χ ′ Σ (G) = ∆ + O(∆ 5 6 ln 1 6 ∆) for a graph G without isolated edges. We improve this upper bound to χ ′ Σ (G) = ∆ + O(∆ 1 2 ) using a simpler approach involving a combinatorial algorithm enhanced by the probabilistic method. The same upper bound is provided for the total version of this problem as well.

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Section: Conjecture 1 ([16]mentioning

confidence: 99%

A note on asymptotically optimal neighbour sum distinguishing colourings

Przybyło

2019

European Journal of Combinatorics

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“…Li et al verified this conjecture for 4 -minor free graphs [6] and planar graphs with the large maximum degree [7]. Wang et al [8] confirmed this conjecture by using the famous Combinatorial Nullstellensatz that holds for any triangle free planar graph with maximum degree of at least 7.…”

Section: Introductionmentioning

confidence: 91%

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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“…The least k which permits constructing such c that attributes distinct weighted degrees to the adjacent vertices in G is called its total neighbour sum distinguishing number and denoted χ ′′ (G), see [38]. Other results concerning this graph invariant can be found in [32,33,38], and in [44], where it has been proved that χ ′′ (G) ≤ ∆(G) + ⌈ 5 3 col(G)⌉. Though constructing a proper total colouring using ∆ plus only a few additional colours is difficult itself, as the long history of the Total Colouring Conjecture exemplifies, we shall prove that asymptotically this many are sufficient to find one satisfying even our additional condition that s(u) = s(v) for every edge uv ∈ E. In particular we shall prove that χ ′′ (G) ≤ (1 + o(1))∆ for all graphs, see Theorem 10 below.…”

Section: Main Objective and Toolsmentioning

confidence: 99%

Asymptotically optimal neighbour sum distinguishing colourings of graphs

Przybyło

2014

Random Struct. Alg.

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Consider a simple graph G = (V,E) and its proper edge colouring c with the elements of the set {1,2,…,k}. The colouring c is said to be neighbour sum distinguishing if for every pair of vertices u, v adjacent in G, the sum of colours of the edges incident with u is distinct from the corresponding sum for v. The smallest integer k for which such colouring exists is known as the neighbour sum distinguishing index of a graph and denoted by χ′∑(G). The definition of this parameter, which makes sense for graphs containing no isolated edges, immediately implies that χ′∑(G)≥Δ, where Δ is the maximum degree of G. On the other hand, it was conjectured by Flandrin et al. that χ′∑(G)≤Δ+2 for all those graphs, except for C5. We prove this bound to be asymptotically correct by showing that χ′∑(G)≤Δ(1+o(1)). The main idea of our argument relays on a random assignment of the colours, where the choice for every edge is biased by so called attractors, randomly assigned to the vertices. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 776–791, 2015

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Neighbor sum distinguishing total colorings of planar graphs

Cited by 56 publications

References 18 publications

A note on asymptotically optimal neighbour sum distinguishing colourings

A note on asymptotically optimal neighbour sum distinguishing colourings

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

Asymptotically optimal neighbour sum distinguishing colourings of graphs

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