ON LIST CHROMATIC NUMBER OF L(2, 0)-LABELING OF A COMPLETE BIPARTITE GRAPH K_{2, n}

2016 ◽  
Vol 99 (10) ◽  
pp. 1571--1582
Author(s):  
Sawitree Ranmechai ◽  
Kittikorn Nakprasit
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Bipartite Graph ◽  
List Chromatic Number

k-Degenerate Graphs

1970 ◽  
Vol 22 (5) ◽  
pp. 1082-1096 ◽  
Author(s):  
Don R. Lick ◽  
Arthur T. White
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Natural Numbers ◽  
Image Position ◽  
Disconnected Graphs ◽  
Totally Disconnected

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

b-Chromatic sum of a graph

2015 ◽  
Vol 07 (04) ◽  
pp. 1550040 ◽  
Author(s):  
P. C. Lisna ◽  
M. S. Sunitha
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Double Star ◽  
Star Graph ◽  
Proper Coloring ◽  
Color Class ◽  
Wheel Graph ◽  
Chromatic Sum

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.

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 A 4-choosable graph that is not (8:2)-choosable

2019 ◽  
Author(s):  
Zdeněk Dvořák ◽  
Xiaolan Hu ◽  
Jean-Sébastien Sereni
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Graph Coloring ◽  
Complete Bipartite Graph ◽  
Line Graph ◽  
Short Paper ◽  
List Coloring ◽  
Proper Coloring ◽  
Famous Theorem ◽  
Seminal Paper

List coloring is a generalization of graph coloring introduced by Erdős, Rubin and Taylor in 1980, which has become extensively studied in graph theory. A graph G is said to be k-choosable, or k-list-colorable, if, for every way of assigning a list (set) of k colors to each vertex of G, it is possible to choose a color from each list in such a way that no two neighboring vertices receive the same color. Note that if the lists are all the same, then this is asking for G to have chromatic number at most k. One might think that the case where all the lists are the same would be the hardest: surely making the lists different should make it easier to ensure that neighboring vertices have different colors. Rather surprisingly, however, this is not the case. A counterexample is provided by the complete bipartite graph K2,4. If the two vertices in the first vertex class are assigned the lists {a,b} and {c,d}, while the vertices in the other vertex class are assigned the lists {a,c}, {a,d}, {b,c} and {b,d}, then it is easy to check that it is not possible to obtain a proper coloring from these lists, so G is not 2-choosable, and yet the chromatic number of G is 2. A famous theorem of Galvin, which solved the so-called Dinitz conjecture, states that the line graph of the complete bipartite graph Kn,n is n-choosable. Equivalently, if each square of an n×n grid is assigned a list of n colors, it is possible to choose a color from each list in such a way that no color appears more than once in any row or column. One can generalize this notion by requiring a choice of not just one color from each list, but some larger number of colors. A graph G is said to be (A,B)-list-colorable if, for every assignment of lists to the vertices of G, each consisting of A colors, there is an assignment of sets of B colors to the vertices such that each vertex is assigned a set that is a subset of its list and the sets assigned to pairs of adjacent vertices are disjoint. (When B=1 this simply says that G is A-choosable.) In this short paper, the authors answer a question that has remained open for almost four decades since it was posed by Erdős, Rubin and Taylor in their seminal paper: if a graph is (A,B)-list-colorable, is it true that it is also (mA,mB)-list-colorable for every m≥1? Quite surprisingly, the answer is again negative - the authors construct a graph that is (4,1)-list-colorable but not (8,2)-list-colorable.

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.

HAJÓS-LIKE THEOREM FOR GROUP COLORING

2010 ◽  
Vol 02 (03) ◽  
pp. 433-436 ◽  
Author(s):  
XINHUI AN ◽  
BAOYINDURENG WU
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Chromatic Number ◽  
Graph Coloring ◽  
Complete Bipartite Graph ◽  
Graph Operations ◽  
Group Coloring

The group coloring of graphs is a new kind of graph coloring, introduced by Jaeger et al. in 1992, and the group chromatic number of a graph G is denoted by χg (G). In this note, we prove that for a positive integer k, a graph G with χg (G)>k can be obtained from any complete bipartite graph G0 with χg(G0)>k by certain types of graph operations.

The coloring of the cozero-divisor graph of a commutative ring

2020 ◽  
Vol 12 (03) ◽  
pp. 2050023
Author(s):  
S. Akbari ◽  
S. Khojasteh
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Local Ring ◽  
Chromatic Number ◽  
Upper Bound ◽  
Complete Bipartite Graph ◽  
Clique Number ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Non Local

Let [Formula: see text] be a commutative ring with unity. The cozero-divisor graph of [Formula: see text] denoted by [Formula: see text] is a graph with the vertex set [Formula: see text], where [Formula: see text] is the set of all nonzero and non-unit elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote the clique number and the chromatic number of [Formula: see text], respectively. In this paper, we prove that if [Formula: see text] is a finite commutative ring, then [Formula: see text] is perfect. Also, we prove that if [Formula: see text] is a commutative Artinian non-local ring and [Formula: see text] is finite, then [Formula: see text]. For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we prove that if [Formula: see text] is a commutative ring, then [Formula: see text] is a complete bipartite graph if and only if [Formula: see text], where [Formula: see text] and [Formula: see text] are fields. Moreover, we present some results on the complete [Formula: see text]-partite cozero-divisor graphs.

scholarly journals b-CHROMATIC NUMBER OF CORONA PRODUCT OF CROWN GRAPH AND COMPLETE BIPARTITE GRAPH WITH PATH GRAPH

2015 ◽  
Vol 2 (2) ◽  
pp. 30-33
Author(s):  
Vijayalakshmi D ◽  
Mohanappriya G
Keyword(s):  
Bipartite Graph ◽  
Structural Properties ◽  
Chromatic Number ◽  
Complete Bipartite Graph ◽  
Proper Coloring ◽  
Path Graph ◽  
Crown Graph ◽  
Corona Product

A b-coloring of a graph is a proper coloring where each color admits at least one node (called dominating node) adjacent to every other used color. The maximum number of colors needed to b-color a graph G is called the b-chromatic number and is denoted by φ(G). In this paper, we find the b-chromatic number and some of the structural properties of corona product of crown graph and complete bipartite graphwith path graph.

scholarly journals Bipartite Graphs whose Squares are not Chromatic-Choosable

10.37236/4343 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Seog-Jin Kim ◽  
Boram Park
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Total Coloring ◽  
Natural Question ◽  
List Total Coloring ◽  
Multipartite Graphs ◽  
Complete Multipartite ◽  
List Chromatic Number ◽  
Total Coloring Conjecture

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $\chi(H)$ and $\chi_{\ell}(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called chromatic-choosable if $\chi_{\ell} (H) = \chi(H)$. It is an interesting problem to find graphs that are chromatic-choosable.Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable for every graph $G$. Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs and are not chromatic choosable. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. Hence a natural question is whether $G^2$ is chromatic-choosable or not for every bipartite graph $G$.In this paper, we give a bipartite graph $G$ such that $\chi_{\ell} (G^2) \neq \chi(G^2)$. Moreover, we show that the value $\chi_{\ell}(G^2) - \chi(G^2)$ can be arbitrarily large.

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling
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