Total-coloring of Sparse Graphs with Maximum Degree 6

2021 ◽  
Vol 37 (4) ◽  
pp. 738-746
Author(s):  
Yu-lin Chang ◽  
Fei Jing ◽  
Guang-hui Wang ◽  
Ji-chang Wu
Keyword(s):  
Maximum Degree ◽  
Total Coloring ◽  
Sparse Graphs

Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 9

2019 ◽  
Vol 342 (5) ◽  
pp. 1392-1402
Author(s):  
Jie Hu ◽  
Guanghui Wang ◽  
Jianliang Wu ◽  
Donglei Yang ◽  
Xiaowei Yu
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring ◽  
Adjacent Vertex

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.

A General Upper Bound on the List Chromatic Number of Locally Sparse Graphs

2002 ◽  
Vol 11 (1) ◽  
pp. 103-111 ◽  
Author(s):  
VAN H. VU
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Present Author ◽  
Upper Bounds ◽  
Constant Factor ◽  
Chromatic Index ◽  
Sparse Graphs ◽  
Critical Graphs ◽  
Multiplicative Constant ◽  
List Chromatic Number

Suppose that G is a graph with maximum degree d(G) such that, for every vertex v in G, the neighbourhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/log f, for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [9], Johansson [7], Alon, Krivelevich and Sudakov [3], and the present author [18]. This also motivates several interesting questions.As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Tuza [6] and Mahdian [13] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [10] concerning the structure of list critical graphs.

Decomposition of sparse graphs into two forests, one having bounded maximum degree

2010 ◽  
Vol 110 (20) ◽  
pp. 913-916 ◽  
Author(s):  
Mickael Montassier ◽  
André Raspaud ◽  
Xuding Zhu
Keyword(s):  
Maximum Degree ◽  
Sparse Graphs

scholarly journals Total coloring of embedded graphs with maximum degree at least seven

2014 ◽  
Vol 518 ◽  
pp. 1-9 ◽  
Author(s):  
Huijuan Wang ◽  
Bin Liu ◽  
Jianliang Wu ◽  
Guizhen Liu
Keyword(s):  
Maximum Degree ◽  
Total Coloring ◽  
Embedded Graphs

scholarly journals The Degree-Diameter Problem for Sparse Graph Classes

10.37236/4313 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Guillermo Pineda-Villavicencio ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Constant Factor ◽  
Average Degree ◽  
Sparse Graph ◽  
Sparse Graphs ◽  
Graph Classes ◽  
Minor Free Graphs

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is $\Theta(\Delta^{\lfloor k/2\rfloor})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the maximum number of vertices up to a constant factor. Other precise bounds are given for graphs embeddable on a given surface and apex-minor-free graphs.

On Brooks' Theorem for Sparse Graphs

1995 ◽  
Vol 4 (2) ◽  
pp. 97-132 ◽  
Author(s):  
Jeong Han Kim
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Sparse Graphs ◽  
Choice Number ◽  
Image Position

Let G be a graph with maximum degree Δ(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfieswhere o(l) goes to zero as Δ(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the list-chromatic (or choice) number:provided g(G) < 4.

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