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.

scholarly journals Face-Degree Bounds for Planar Critical Graphs

10.37236/5895 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Ligang Jin ◽  
Yingli Kang ◽  
Eckhard Steffen
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Upper Bounds ◽  
Partial Characterization ◽  
Critical Graphs ◽  
Degree Bounds

The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs,  based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters for smallest counterexamples to Vizing's conjecture, thus providing a partial characterization of such graphs, if they exist.For $k \leq 5$ the results give insights into the structure of planar edge-chromatic critical graphs.

Coloring squares of graphs via vertex orderings

2020 ◽  
pp. 2050093
Author(s):  
Mehmet Akif Yetim
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Chordal Graph ◽  
Upper Bounds ◽  
Cocomparability Graph ◽  
Vertex Ordering

We provide upper bounds on the chromatic number of the square of graphs, which have vertex ordering characterizations. We prove that [Formula: see text] is [Formula: see text]-colorable when [Formula: see text] is a cocomparability graph, [Formula: see text]-colorable when [Formula: see text] is a strongly orderable graph and [Formula: see text]-colorable when [Formula: see text] is a dually chordal graph, where [Formula: see text] is the maximum degree and [Formula: see text] = max[Formula: see text] is the multiplicity of the graph [Formula: see text]. This improves the currently known upper bounds on the chromatic number of squares of graphs from these classes.

scholarly journals Star multigraphs with three vertices of maximum degree

1986 ◽  
Vol 100 (2) ◽  
pp. 303-317 ◽  
Author(s):  
A. G. Chetwynd ◽  
A. J. W. Hilton
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Special Kind ◽  
Chromatic Index ◽  
Simple Graphs ◽  
Multiple Edges ◽  
Image Position ◽  
Edge Colouring

The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with each non-simple edge. An edge-colouring of a multigraph G is a map ø: E(G)→, where is a set of colours and E(G) is the set of edges of G, such that no two edges receiving the same colour have a vertex in common. The chromatic index, or edge-chromatic numberχ′(G) of G is the least value of || for which an edge-colouring of G exists. Generalizing a well-known theorem of Vizing [14], we showed in [6] that, for a star-multigraph G,where Δ(G) denotes the maximum degree (that is, the maximum number of edges incident with a vertex) of G. Star-multigraphs for which χ′(G) = Δ(G) are said to be Class 1, and otherwise they are Class 2.

Upper Bounds on List Star Chromatic Index of Sparse Graphs

2019 ◽  
Vol 36 (1) ◽  
pp. 1-12
Author(s):  
Jia Ao Li ◽  
Katie Horacek ◽  
Rong Luo ◽  
Zheng Ke Miao
Keyword(s):  
Upper Bounds ◽  
Chromatic Index ◽  
Sparse 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.

scholarly journals The Distance-t Chromatic Index of Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 90-101 ◽  
Author(s):  
TOMÁŠ KAISER ◽  
ROSS J. KANG
Keyword(s):  
Maximum Degree ◽  
Upper Bounds ◽  
The Other ◽  
Graph Colouring ◽  
Chromatic Index ◽  
Multiplicative Factor ◽  
Strong Chromatic Index ◽  
Absolute Positive Constant ◽  
Large Maximum ◽  
Fixed Positive Integer

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-ε)Δt for graphs of maximum degree at most Δ, where ε is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

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.

scholarly journals Orthogonal Colorings of Graphs

10.37236/1437 ◽  
1998 ◽  
Vol 6 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Latin Squares ◽  
Upper Bounds ◽  
Strong Version ◽  
Orthogonal Latin Squares ◽  
Graph Parameters

An orthogonal coloring of a graph $G$ is a pair $\{c_1,c_2\}$ of proper colorings of $G$, having the property that if two vertices are colored with the same color in $c_1$, then they must have distinct colors in $c_2$. The notion of orthogonal colorings is strongly related to the notion of orthogonal Latin squares. The orthogonal chromatic number of $G$, denoted by $O\chi(G)$, is the minimum possible number of colors used in an orthogonal coloring of $G$. If $G$ has $n$ vertices, then the definition implies that $\left\lceil \sqrt{n} \, \right\rceil \leq O\chi(G) \leq n$. $G$ is said to have an optimal orthogonal coloring if $O\chi(G) = \left\lceil \sqrt{n} \, \right\rceil$. If, in addition, $n$ is an integer square, then we say that $G$ has a perfect orthogonal coloring, since for any two colors $x$ and $y$, there is exactly one vertex colored by $x$ in $c_1$ and by $y$ in $c_2$. The purpose of this paper is to study the parameter $O\chi(G)$ and supply upper bounds to it which depend on other graph parameters such as the maximum degree and the chromatic number. We also study the structure of graphs having an optimal or perfect orthogonal coloring, and show that several classes of graphs always have an optimal or perfect orthogonal coloring. We also consider the strong version of orthogonal colorings, where no vertex may receive the same color in both colorings.

scholarly journals On-line List Colouring of Random Graphs

10.37236/5003 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Alan Frieze ◽  
Dieter Mitsche ◽  
Xavier Pérez-Giménez ◽  
Paweł Prałat
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Constant Factor ◽  
Average Degree ◽  
Multiplicative Constant ◽  
Line List ◽  
Multiplicative Factor ◽  
On Line ◽  
Choice Number

In this paper, the on-line list colouring of binomial random graphs $\mathcal{G}(n,p)$ is studied. We show that the on-line choice number of $\mathcal{G}(n,p)$ is asymptotically almost surely asymptotic to the chromatic number of $\mathcal{G}(n,p)$, provided that the average degree $d=p(n-1)$ tends to infinity faster than $(\log \log n)^{1/3} (\log n)^2 n^{2/3}$. For sparser graphs, we are slightly less successful; we show that if $d \ge (\log n)^{2+\epsilon}$ for some $\epsilon>0$, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of $C$, where $C \in [2,4]$, depending on the range of $d$. Also, for $d=O(1)$, the on-line choice number is by at most a multiplicative constant factor larger than the chromatic number.

Linear list r-hued coloring of sparse graphs

2018 ◽  
Vol 10 (04) ◽  
pp. 1850045
Author(s):  
Hongping Ma ◽  
Xiaoxue Hu ◽  
Jiangxu Kong ◽  
Murong Xu
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Average Degree ◽  
Disjoint Paths ◽  
Proper Coloring ◽  
Sparse Graphs ◽  
Maximum Average Degree ◽  
Linear Formula ◽  
Linear List

An [Formula: see text]-hued coloring is a proper coloring such that the number of colors used by the neighbors of [Formula: see text] is at least [Formula: see text]. A linear [Formula: see text]-hued coloring is an [Formula: see text]-hued coloring such that each pair of color classes induces a union of disjoint paths. We study the linear list [Formula: see text]-hued chromatic number, denoted by [Formula: see text], of sparse graphs. It is clear that any graph [Formula: see text] with maximum degree [Formula: see text] satisfies [Formula: see text]. Let [Formula: see text] be the maximum average degree of a graph [Formula: see text]. In this paper, we obtain the following results: (1) If [Formula: see text], then [Formula: see text] (2) If [Formula: see text], then [Formula: see text]. (3) If [Formula: see text], then [Formula: see text].

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