Total chromatic number of planar graphs with maximum degree ten

2006 ◽  
Vol 54 (2) ◽  
pp. 91-102 ◽  
Author(s):  
Weifan Wang
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Total Chromatic Number

scholarly journals Total Colorings of $F_5$-Free Planar Graphs with Maximum Degree 8

10.37236/3303 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Jian Chang ◽  
Jian-Liang Wu ◽  
Hui-Juan Wang ◽  
Zhan-Hai Guo
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Fan Graph

The total chromatic number of a graph $G$, denoted by $\chi′′(G)$, is the minimum number of colors needed to color the vertices and edges of $G$ such that no two adjacent or incident elements get the same color. It is known that if a planar graph $G$ has maximum degree $\Delta ≥ 9$, then $\chi′′(G) = \Delta + 1$. The join $K_1 \vee P_n$ of $K_1$ and $P_n$ is called a fan graph $F_n$. In this paper, we prove that if $G$ is a $F_5$-free planar graph with maximum degree 8, then $\chi′′(G) = 9$.

Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10

2017 ◽  
Vol 314 ◽  
pp. 456-468 ◽  
Author(s):  
Donglei Yang ◽  
Lin Sun ◽  
Xiaowei Yu ◽  
Jianliang Wu ◽  
Shan Zhou
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Total Chromatic Number

The structure of plane graphs with independent crossings and its applications to coloring problems

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Xin Zhang ◽  
Guizhen Liu
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Plane Graph ◽  
Plane Graphs ◽  
Total Chromatic Number ◽  
Linear Arboricity

AbstractIf a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity of G is ℈Δ/2⌊ if Δ ≥ 17, where G is an IC-planar graph and Δ is the maximum degree of G.

scholarly journals Total Coloring Conjecture for Certain Classes of Graphs

Algorithms ◽  
2018 ◽  
Vol 11 (10) ◽  
pp. 161 ◽  
Author(s):  
R. Vignesh ◽  
J. Geetha ◽  
K. Somasundaram
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Product Line ◽  
Total Coloring ◽  
Lexicographic Product ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Total Coloring Conjecture

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

Total colorings of circulant graphs

2020 ◽  
pp. 2150050
Author(s):  
J. Geetha ◽  
K. Somasundaram ◽  
Hung-Lin Fu
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Total Coloring ◽  
Circulant Graphs ◽  
Total Chromatic Number ◽  
Number Formula ◽  
Total Coloring Conjecture

The total chromatic number [Formula: see text] is the least number of colors needed to color the vertices and edges of a graph [Formula: see text] such that no incident or adjacent elements (vertices or edges) receive the same color. Behzad and Vizing proposed a well-known total coloring conjecture (TCC): [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. For the powers of cycles, Campos and de Mello proposed the following conjecture: Let [Formula: see text] denote the graphs of powers of cycles of order [Formula: see text] and length [Formula: see text] with [Formula: see text]. Then, [Formula: see text] In this paper, we prove the Campos and de Mello’s conjecture for some classes of powers of cycles. Also, we prove the TCC for complement of powers of cycles.

List injective coloring of planar graphs with disjoint 5−-cycles

2021 ◽  
pp. 2150148
Author(s):  
Wenwen Li ◽  
Jiansheng Cai
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Injective Coloring

An injective [Formula: see text]-coloring of a graph [Formula: see text] is called injective if any two vertices joined by a path of length two get different colors. A graph [Formula: see text] is injectively [Formula: see text]-choosable if for any color list [Formula: see text] of admissible colors on [Formula: see text] of size [Formula: see text] it allows an injective coloring [Formula: see text] such that [Formula: see text] whenever [Formula: see text]. Let [Formula: see text], [Formula: see text] denote the injective chromatic number and injective choosability number of [Formula: see text], respectively. Let [Formula: see text] be a plane with disjoint [Formula: see text]-cycles and maximum degree [Formula: see text]. We show that [Formula: see text] if [Formula: see text], then [Formula: see text]; [Formula: see text] if [Formula: see text], then [Formula: see text].

LIST INJECTIVE COLORING OF PLANAR GRAPHS WITH GIRTH g ≥ 5

2014 ◽  
Vol 06 (01) ◽  
pp. 1450006 ◽  
Author(s):  
YUEHUA BU ◽  
SHENG YANG
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Plane Graph ◽  
Common Neighbor ◽  
Injective Coloring

An injective-k coloring of a graph G is a mapping cV(G) → {1, 2, …, k}, such that c(u) ≠ c(v) for each u, v ∈ V(G), whenever u, v have a common neighbor in G. If G has an injective-k coloring, then we call that G is injective-k colorable. Call χi(G) = min {k | G is injective-k colorable} is the injective chromatic number of G. Assign each vertex v ∈ V(G) a coloring set L(v), then L = {L(v) | v ∈ V(G)} is said to be a color list of G. Let L be a color list of G, if G has an injective coloring c such that c(v) ∈ L(v), ∀v ∈ V(G), then we call c an injective L-coloring of G. If for any color list L, such that |L(v)| ≥ k, G has an injective L-coloring, then G is said to be injective k-choosable. Call [Formula: see text] is injective k-choosable} is the injective chromatic number of G. So far, for the plane graph G of girth g(G) ≥ 5 and maximum degree Δ(G) ≥ 8, the best result of injective chromatic number is χi(G) ≤ Δ + 8. In this paper, for the plane graph G, we proved that [Formula: see text] if girth g(G) ≥ 5 and maximum degree Δ(G) ≥ 8.

2-distance vertex-distinguishing total coloring of graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850018
Author(s):  
Yafang Hu ◽  
Weifan Wang
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Number Formula ◽  
Proper Total Coloring ◽  
Distance Formula

A [Formula: see text]-distance vertex-distinguishing total coloring of a graph [Formula: see text] is a proper total coloring of [Formula: see text] such that any pair of vertices at distance [Formula: see text] have distinct sets of colors. The [Formula: see text]-distance vertex-distinguishing total chromatic number [Formula: see text] of [Formula: see text] is the minimum number of colors needed for a [Formula: see text]-distance vertex-distinguishing total coloring of [Formula: see text]. In this paper, we determine the [Formula: see text]-distance vertex-distinguishing total chromatic number of some graphs such as paths, cycles, wheels, trees, unicycle graphs, [Formula: see text], and [Formula: see text]. We conjecture that every simple graph [Formula: see text] with maximum degree [Formula: see text] satisfies [Formula: see text].

scholarly journals Total chromatic number of graphs of high degree, II

1992 ◽  
Vol 53 (2) ◽  
pp. 219-228 ◽  
Author(s):  
H. P. Yap ◽  
K. H. Chew
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Independent Set ◽  
Simple Graph ◽  
Simple Graphs ◽  
Total Chromatic Number ◽  
Prove Theorem ◽  
High Degree

AbstractWe prove Theorem 1: suppose G is a simple graph of order n having Δ(G) = n − k where k ≥ 5 and n ≥ max (13, 3k −3). If G contains an independent set of k − 3 vertices, then the TCC (Total Colouring Conjecture) is true. Applying Theorem 1, we also prove that the TCC is true for any simple graph G of order n having Δ(G) = n −5. The latter result together with some earlier results confirm that the TCC is true for all simple graphs whose maximum degree is at most four and for all simple graphs of order n having maximum degree at least n − 5.

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