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11 pages, 274 KB

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 265

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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11 pages, 354 KB

Open AccessArticle

Equitable Coloring of IC-Planar Graphs with Girth g ≥ 7

by Danjun Huang and Xianxi Wu

Axioms 2023, 12(9), 822; https://doi.org/10.3390/axioms12090822 - 26 Aug 2023

Viewed by 1761

Abstract

An equitable k-coloring of a graph G is a proper vertex coloring such that the size of any two color classes differ at most 1. If there is an equitable k-coloring of G, then the graph G is said to be equitably k-colorable. A 1-planar graph is a graph that can be embedded in the Euclidean plane such that each edge can be crossed by other edges at most once. An IC-planar graph is a 1-planar graph with distinct end vertices of any two crossings. In this paper, we will prove that every IC-planar graph with girth

g \geq 7

is equitably

Δ (G)

-colorable, where

Δ (G)

is the maximum degree of G. Full article

(This article belongs to the Special Issue Recent Advances in Graph Theory with Applications)

9 pages, 386 KB

Open AccessArticle

The Surviving Rate of IC-Planar Graphs

by Xiaoxue Hu, Jiacheng Hu and Jiangxu Kong

Symmetry 2022, 14(6), 1258; https://doi.org/10.3390/sym14061258 - 17 Jun 2022

Viewed by 2364

Abstract

Let k and n be two positive integers. Firefighting is a discrete dynamical process of preventing the spread of fire. Let G be a connected graph G with n vertices. Assuming a fire starts at one of the vertices of G, the firefighters choose k unburned vertices at each step, and then the fire spreads to all unprotected neighbors of the burning vertices. The process continues until the fire stops spreading. The goal is to protect as many vertices as possible. When a fire breaks out randomly at a vertex of G, its k-surviving rate,

ρ_{k} (G)

, is the expected number of saved vertices. A graph is IC-planar if it has a drawing in which each edge cross once and their endpoints are disjoint. In this paper, we prove that

ρ_{4} (G) > \frac{1}{124}

for every IC-planar graph G. This is proven by the discharging method and the locally symmetric of the graph. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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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 2106

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 2276

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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