scholarly journals The Connectivity of a Bipartite Graph and Its Bipartite Complementary Graph

2020 ◽  
Vol 30 (03) ◽  
pp. 2040005
Author(s):  
Yingzhi Tian ◽  
Huaping Ma ◽  
Liyun Wu
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Type Inequality ◽  
Bipartite Graphs ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Complementary Graph

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph [Formula: see text] and the same invariant in the complement [Formula: see text] of [Formula: see text] is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph [Formula: see text] with bipartition ([Formula: see text]), its bipartite complementary graph [Formula: see text] is a bipartite graph with [Formula: see text] and [Formula: see text] and [Formula: see text]. In this paper, we obtain the Nordhaus-Gaddum type inequalities for connectivity of bipartite graphs and its bipartite complementary graphs. Furthermore, we prove that these inequalities are best possible.

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.

scholarly journals Harary index of the k-th power of a graph

2013 ◽  
Vol 7 (1) ◽  
pp. 94-105 ◽  
Author(s):  
Guifu Su ◽  
Liming Xiong ◽  
Ivan Gutman
Keyword(s):  
Type Inequality ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Harary Index ◽  
Underlying Graph

The k-th power of a graph G, denoted by Gk, is a graph with the same set of vertices as G, such that two vertices are adjacent in Gk if and only if their distance in G is at most k. The Harary index H is the sum of the reciprocal distances of all pairs of vertices of the underlying graph. Lower and upper bounds on H(Gk) are obtained. A Nordhaus-Gaddum type inequality for H(Gk) is also established.

scholarly journals Estimating vertex-degree-based energies

2022 ◽  
Vol 70 (1) ◽  
pp. 13-23
Author(s):  
Ivan Gutman
Keyword(s):  
General Theory ◽  
Spectral Theory ◽  
Current Literature ◽  
Bipartite Graphs ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Lower And Upper Bounds ◽  
Graph Invariants ◽  
General Properties ◽  
Graph Energy

Introduction/purpose: In the current literature, several dozens of vertex-degree-based (VDB) graph invariants are being studied. To each such invariant, a matrix can be associated. The VDB energy is the energy (= sum of the absolute values of the eigenvalues) of the respective VDB matrix. The paper examines some general properties of the VDB energy of bipartite graphs. Results: Estimates (lower and upper bounds) are established for the VDB energy of bipartite graphs in which there are no cycles of size divisible by 4, in terms of ordinary graph energy. Conclusion: The results of the paper contribute to the spectral theory of VDB matrices, especially to the general theory of VDB energy.

Algorithmic Aspects of Partial List Colourings

2000 ◽  
Vol 9 (4) ◽  
pp. 375-380 ◽  
Author(s):  
M. VOIGT
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Polynomial Time ◽  
Bipartite Graphs ◽  
Partial List ◽  
List Chromatic Number

Let G = (V, E) be a graph with n vertices, chromatic number χ(G) and list chromatic number χ[lscr ](G). Suppose each vertex of V(G) is assigned a list of t colours. Albertson, Grossman and Haas [1] conjectured that at least [formula here] vertices can be coloured properly from these lists.Albertson, Grossman and Haas [1] and Chappell [3] proved partial results concerning this conjecture. This paper presents algorithms that colour at least the number of vertices given in the bounds of Albertson, Grossman and Haas, and Chappell. In particular, it follows that the conjecture is valid for all bipartite graphs and that, for every bipartite graph and every assignment of lists with t colours in each list where 0 [les ] t [les ] χ[lscr ](G), it is possible to colour at least (1 − (1/2)t)n vertices in polynomial time. Thus, if G is bipartite and [Lscr ] is a list assignment with [mid ]L(v)[mid ] [ges ] log2n for all v ∈ V, then G is [Lscr ]-list colourable in polynomial time.

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 $\mathcal M$ .

scholarly journals Some Extremal Graphs with Respect to Sombor Index

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1202
Author(s):  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Molecular Structure ◽  
Chromatic Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Graph Parameters ◽  
Structure Descriptor

Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge set E(G). Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) and is defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of the vertex vi in G. In this paper we present some lower and upper bounds on the Sombor index of graph G in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs.

scholarly journals Results on a Chromatic Number of a Bi-polar Fuzzy Complete Bipartite Graphs and Labelling of Tri-Polar Fuzzy Graphs

2021 ◽  
Vol 12 (2) ◽  
pp. 1040-1046
Author(s):  
Remala Mounika Lakshmi, Et. al.
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Bipartite Graph ◽  
Research Work ◽  
Bipartite Graphs ◽  
Network System ◽  
Route Network ◽  
Fuzzy Graphs ◽  
Complete Bipartite Graphs ◽  
Ultimate Objective

The ultimate objective of a piece of research work is to present the labelling of vertices in 3-PFG and labelling of distances in 3-PFG. Also, we characterize some of its properties. Later, we define the vertex and edge chromatic number BF- Complete Bipartite graph. Further we illustrated an example for BFRGS which represents a Route Network system.

scholarly journals The Game Chromatic Number of Dense Random Graphs

10.37236/4391 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Ralph Keusch ◽  
Angelika Steger
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
High Probability ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Game Chromatic Number ◽  
Order Of Magnitude ◽  
Random Structures

Suppose that two players take turns coloring the vertices of a given graph G with k colors. In each move the current player colors a vertex such that neighboring vertices get different colors. The first player wins this game if and only if at the end, all vertices are colored. The game chromatic number χg(G) is defined as the smallest k for which the first player has a winning strategy.Recently, Bohman, Frieze and Sudakov [Random Structures and Algorithms 2008] analysed the game chromatic number of random graphs and obtained lower and upper bounds of the same order of magnitude. In this paper we improve existing results and show that with high probability, the game chromatic number χg(Gn,p) of dense random graphs with p ≥ e-o(log n) is asymptotically twice as large as the ordinary chromatic number χ(Gn,p).

scholarly journals Distinguishing Chromatic Numbers of Bipartite Graphs

10.37236/165 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
C. Laflamme ◽  
K. Seyffarth
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Chromatic Numbers

Extending the work of K.L. Collins and A.N. Trenk, we characterize connected bipartite graphs with large distinguishing chromatic number. In particular, if $G$ is a connected bipartite graph with maximum degree $\Delta \geq 3$, then $\chi_D(G)\leq 2\Delta -2$ whenever $G\not\cong K_{\Delta-1,\Delta}$, $K_{\Delta,\Delta}$.

scholarly journals Locating-chromatic number of the edge-amalgamation of trees

2020 ◽  
Vol 4 (2) ◽  
pp. 126
Author(s):  
Dian Kastika Syofyan ◽  
Edy Tri Baskoro ◽  
Hilda Assiyatun
Keyword(s):  
Chromatic Number ◽  
Upper Bounds ◽  
Title Page ◽  
Metric Dimension ◽  
Chromatic Numbers ◽  
Lower And Upper Bounds ◽  
Vertex Set ◽  
Partition Dimension ◽  
Special Case

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The investigation on the locating-chromatic number of a graph was initiated by Chartrand </span><span>et al. </span><span>(2002). This concept is in fact a special case of the partition dimension of a graph. This topic has received much attention. However, the results are still far from satisfaction. We can define the locating-chromatic number of a graph </span><span>G </span><span>as the smallest integer </span><span>k </span><span>such that there exists a </span><span>k</span><span>-partition of the vertex-set of </span><span>G </span><span>such that all vertices have distinct coordinates with respect to this partition. As we know that the metric dimension of a tree is completely solved. However, the locating-chromatic numbers for most of trees are still open. For </span><span><em>i</em> </span><span>= 1</span><span>, </span><span>2</span><span>, . . . , <em>t</em>, </span><span>let </span><em>T</em><span>i </span><span>be a tree with a fixed edge </span><span>e</span><span>o</span><span>i </span><span>called the terminal edge. The edge-amalgamation of all </span><span>T</span><span>i</span><span>s </span><span>denoted by Edge-Amal</span><span>{</span><span>T</span><span>i</span><span>;</span><span>e</span><span>o</span><span>i</span><span>} </span><span>is a tree formed by taking all the </span><span>T</span><span>i</span><span>s and identifying their terminal edges. In this paper, we study the locating-chromatic number of the edge-amalgamation of arbitrary trees. We give lower and upper bounds for their locating-chromatic numbers and show that the bounds are tight.</span></p></div></div></div>

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