A Unique Approach on Upper Bounds for the Chromatic Number of Total Graphs

2012 ◽  
Vol 5 (4) ◽  
pp. 240-246
Author(s):  
J. Venkateswara Rao ◽  
R.V.N. Srinivasa Rao
Keyword(s):  
Chromatic Number ◽  
Upper Bounds ◽  
Unique Approach

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.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

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.

Graph coloring approach with new upper bounds for the chromatic number: team building application

2018 ◽  
Vol 52 (3) ◽  
pp. 807-818
Author(s):  
Assia Gueham ◽  
Anass Nagih ◽  
Hacene Ait Haddadene ◽  
Malek Masmoudi
Keyword(s):  
Chromatic Number ◽  
Graph Coloring ◽  
Upper Bound ◽  
Team Building ◽  
Upper Bounds ◽  
Real Case ◽  
Soft Constraints ◽  
Empirical Comparison ◽  
Hard And Soft Constraints

In this paper, we focus on the coloration approach and estimation of chromatic number. First, we propose an upper bound of the chromatic number based on the orientation algorithm described in previous studies. This upper bound is further improved by developing a novel coloration algorithm. Second, we make a theoretical and empirical comparison of our bounds with Brooks’s bound and Reed’s conjecture for class of triangle-free graphs. Third, we propose an adaptation of our algorithm to deal with the team building problem respecting several hard and soft constraints. Finally, a real case study from healthcare domain is considered for illustration.

Improved upper bounds for nearly antipodal chromatic number of paths

2007 ◽  
Vol 27 (1) ◽  
pp. 159
Author(s):  
Wen-Jie He ◽  
Yu-Fa Shen ◽  
Guo-Ping Zheng
Keyword(s):  
Chromatic Number ◽  
Upper Bounds

scholarly journals Three new upper bounds on the chromatic number

2011 ◽  
Vol 159 (18) ◽  
pp. 2281-2289 ◽  
Author(s):  
María Soto ◽  
André Rossi ◽  
Marc Sevaux
Keyword(s):  
Chromatic Number ◽  
Upper Bounds

scholarly journals Bounds on the Distinguishing Chromatic Number

10.37236/177 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Karen L. Collins ◽  
Mark Hovey ◽  
Ann N. Trenk
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Automorphism Group ◽  
Chromatic Number ◽  
Automorphism Groups ◽  
Upper Bounds ◽  
Underlying Graph ◽  
Minimum Number

Collins and Trenk define the distinguishing chromatic number $\chi_D(G)$ of a graph $G$ to be the minimum number of colors needed to properly color the vertices of $G$ so that the only automorphism of $G$ that preserves colors is the identity. They prove results about $\chi_D(G)$ based on the underlying graph $G$. In this paper we prove results that relate $\chi_D(G)$ to the automorphism group of $G$. We prove two upper bounds for $\chi_D(G)$ in terms of the chromatic number $\chi(G)$ and show that each result is tight: (1) if Aut$(G)$ is any finite group of order $p_1^{i_1} p_2^{i_2} \cdots p_k^{i_k}$ then $\chi_D(G) \le \chi(G) + i_1 + i_2 \cdots + i_k$, and (2) if Aut$(G)$ is a finite and abelian group written Aut$(G) = {\Bbb Z}_{p_{1}^{i_{1}}}\times \cdots \times {\Bbb Z}_{p_{k}^{i_{k}}}$ then we get the improved bound $\chi_D(G) \le \chi(G) + k$. In addition, we characterize automorphism groups of graphs with $\chi_D(G) = 2$ and discuss similar results for graphs with $\chi_D(G)=3$.

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 All finite sets are Ramsey in the maximum norm

2021 ◽  
Vol 9 ◽  
Author(s):  
Andrey Kupavskii ◽  
Arsenii Sagdeev
Keyword(s):  
Chromatic Number ◽  
Metric Space ◽  
Metric Spaces ◽  
Upper Bounds ◽  
Maximum Norm ◽  
Lower And Upper Bounds ◽  
Finite Sets ◽  
Finite Metric Space ◽  
Monochromatic Copy

Abstract For two metric spaces $\mathbb X$ and $\mathcal Y$ the chromatic number $\chi ({{\mathbb X}};{{\mathcal{Y}}})$ of $\mathbb X$ with forbidden $\mathcal Y$ is the smallest k such that there is a colouring of the points of $\mathbb X$ with k colors that contains no monochromatic copy of $\mathcal Y$ . In this article, we show that for each finite metric space $\mathcal {M}$ that contains at least two points the value $\chi \left ({{\mathbb R}}^n_\infty; \mathcal M \right )$ grows exponentially with n. We also provide explicit lower and upper bounds for some special $\mathcal M$ .

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