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Open AccessArticle

Neighbor Sum Distinguishing Total Choice Number of IC-Planar Graphs Without 4-Cycles

by Meili Ye and Donghan Zhang

Mathematics 2026, 14(10), 1663; https://doi.org/10.3390/math14101663 - 13 May 2026

Viewed by 157

A neighbor sum distinguishing (NSD) total coloring of a graph G is a mapping

ϕ : T (G) = V (G) \cup E (G) \to {1, 2, \dots, k}

such that any [...] Read more.

A neighbor sum distinguishing (NSD) total coloring of a graph G is a mapping

ϕ : T (G) = V (G) \cup E (G) \to {1, 2, \dots, k}

such that any two adjacent or incident elements in

T (G)

receive different colors, and the sum of the colors of all incident edges of u and the color of u is different from the sum of the colors of all incident edges of v and the color of v for each edge

u v

. The NSD total chromatic number of G, denoted by

χ_{Σ}^{t} (G)

, is the smallest integer k such that G has an NSD total coloring. For any graph G, there is a conjecture that the NSD total chromatic number

χ_{Σ}^{t} (G) \leq Δ (G) + 3

, where

Δ (G)

denotes the maximum degree of G. The neighbor sum distinguishing total choice number of G, denoted by

{ch}_{Σ}^{t} (G)

, is the smallest integer k such that, after assigning each

z \in T (G)

a set

L (z)

of k real numbers, G has an NSD total coloring

ϕ

satisfying

ϕ (z) \in L (z)

for each

z \in T (G)

. Obviously,

χ_{Σ}^{t} (G) \leq

{ch}_{Σ}^{t} (G)

. In this paper, we prove that

{ch}_{Σ}^{t} (G) \leq Δ (G) + 3

for any IC-planar graph G without 4-cycles and

Δ (G) \geq 7

by applying the Combinatorial Nullstellensatz, which improves upon the previous results. Full article

(This article belongs to the Section E: Applied Mathematics)

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13 pages, 290 KB

Open AccessArticle

On the Total Neighbor Sum Distinguishing Index of IC-Planar Graphs

by Donghan Zhang, Chao Li and Fugang Chao

Symmetry 2021, 13(10), 1787; https://doi.org/10.3390/sym13101787 - 26 Sep 2021

Cited by 3 | Viewed by 2076

Abstract

A proper total k-coloring

ϕ

of G with

\sum_{z \in E_{G} (u) \cup {u}} ϕ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} ϕ (z)

for [...] Read more.

A proper total k-coloring

ϕ

of G with

\sum_{z \in E_{G} (u) \cup {u}} ϕ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} ϕ (z)

for each

u v \in E (G)

is called a total neighbor sum distinguishing k-coloring, where

E_{G} (u) = {u v | u v \in E (G)}

. Pilśniak and Woźniak conjectured that every graph with maximum degree

Δ

exists a total neighbor sum distinguishing

(Δ + 3)

-coloring. In this paper, we proved that any IC-planar graph with

Δ \geq 12

satisfies this conjecture, which improves the result of Song and Xu. Full article

(This article belongs to the Special Issue Research on Symmetry Applied in Graph Theory)

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11 pages, 285 KB

Open AccessArticle

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ_2,1,2

by Donghan Zhang

Mathematics 2021, 9(7), 708; https://doi.org/10.3390/math9070708 - 25 Mar 2021

Cited by 12 | Viewed by 2249

Abstract

A theta graph

Θ_{2, 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 [...] Read more.

A theta graph

Θ_{2, 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

\sum_{z \in E_{G} (u) \cup {u}} ϕ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} ϕ (z)

for each edge

u v \in E (G)

, where

E_{G} (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 \geq 9

but without theta graphs

Θ_{2, 1, 2}

by applying the Combinatorial Nullstellensatz, which improves the result of Song et al. Full article

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