scholarly journals On r-dynamic coloring of comb graphs

2021 ◽  
Vol 27 (2) ◽  
pp. 191-200
Author(s):  
K. Kalaiselvi ◽  
◽  
N. Mohanapriya ◽  
J. Vernold Vivin ◽  
◽  
...  
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Middle Graph ◽  
Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

scholarly journals ON DYNAMIC COLORING OF WEB GRAPH

2018 ◽  
Vol 5 (2) ◽  
pp. 11-15
Author(s):  
Aaresh R.R ◽  
Venkatachalam M ◽  
Deepa T
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Web Graph ◽  
Middle Graph

Dynamic coloring of a graph G is a proper coloring. The chromatic number of a graph G is the minimum k such that G has a dynamic coloring with k colors. In this paper we investigate the dynamic chromatic number for the Central graph, Middle graph, Total graph and Line graph of Web graph Wn denoted by C(Wn), M(Wn), T(Wn) and L(Wn) respectively.

On b-chromatic number of theta graph families

2020 ◽  
pp. 643-650
Author(s):  
M. Vinitha ◽  
M. Venkatachalam
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Total Graph ◽  
Middle Graph ◽  
Graph Families

In this paper, we investigate the $b$-chromatic number for the theta graph $\theta(s_1, s_2, \cdots, s_n)$, middle graph of theta graph $M(G)$, total graph of theta graph $T(G)$, line graph of theta graph $L(G)$ and the central graph of theta graph $C(G)$.

scholarly journals On the r-dynamic coloring of some fan graph families

2021 ◽  
Vol 29 (3) ◽  
pp. 151-181
Author(s):  
Raúl M. Falcón ◽  
M. Venkatachalam ◽  
S. Gowri ◽  
G. Nandini
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Graph Invariant ◽  
Total Graph ◽  
Sharp Bounds ◽  
Middle Graph ◽  
Graph Families ◽  
Fan Graph

Abstract In this paper, we determine the r-dynamic chromatic number of the fan graph Fm,n and determine sharp bounds of this graph invariant for four related families of graphs: The middle graph M(Fm,n ), the total graph T (Fm,n ), the central graph C(Fm,n ) and the line graph L(Fm,n ). In addition, we determine the r-dynamic chromatic number of each one of these four families of graphs in case of being m = 1.

scholarly journals Local chromatic number and topology

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gábor Simonyi ◽  
Gábor Tardos
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Topological Method ◽  
Proper Coloring ◽  
And Topology ◽  
Minimum Number ◽  
International Audience ◽  
Closed Neighborhood

International audience The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.

scholarly journals Colorful Subhypergraphs in Kneser Hypergraphs

10.37236/3573 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Frédéric Meunier
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Uniform Hypergraph ◽  
Proper Coloring ◽  
Kneser Graph ◽  
Complete Bipartite Graphs ◽  
Definition Of ◽  
Ky Fan

Using a $\mathbb{Z}_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a uniform hypergraph).

scholarly journals Induced Ramsey Number for a Star Versus a Fixed Graph

10.37236/9358 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Maria Axenovich ◽  
Izolda Gorgol
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Constant Factor ◽  
Ramsey Number ◽  
Large N ◽  
Star Versus

We write $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$ for graphs $F, G,$ and $H$, if for any coloring of the edges of $F$ in red and blue, there is either a red induced copy of $H$ or a blue induced copy of $G$. For graphs $G$ and $H$, let $\mathrm{IR}(H,G)$ be the smallest number of vertices in a graph $F$ such that $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$. In this note we consider the case when $G$ is a star on $n$ edges, for large $n$ and $H$ is a fixed graph. We prove that  $$ (\chi(H)-1) n \leq \mathrm{IR}(H, K_{1,n}) \leq (\chi(H)-1)^2n + \epsilon n,$$ for any $\epsilon>0$,  sufficiently large $n$, and $\chi(H)$ denoting the chromatic number of $H$. The lower bound is asymptotically tight  for any fixed bipartite $H$. The upper bound is attained up to a constant factor, for example when $H$ is a clique.

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 Off-Diagonal Ordered Ramsey Numbers of Matchings

10.37236/8085 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Dhruv Rohatgi
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Relative Order ◽  
Ramsey Numbers ◽  
Asymptotic Bounds ◽  
Special Cases ◽  
Ordered Graphs

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete ordered graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the off-diagonal ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that there is some fixed $\epsilon > 0$ such that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for ordered matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random ordered matchings with interval chromatic number $2$.

A note on global dominator coloring of graphs

2021 ◽  
pp. 2150158
Author(s):  
R. Rangarajan ◽  
David. A. Kalarkop
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Proper Coloring ◽  
Color Class ◽  
Minimum Number ◽  
Even Order ◽  
Dominator Coloring

Global dominator coloring of the graph [Formula: see text] is the proper coloring of [Formula: see text] such that every vertex of [Formula: see text] dominates atleast one color class as well as anti-dominates atleast one color class. The minimum number of colors required for global dominator coloring of [Formula: see text] is called global dominator chromatic number of [Formula: see text] denoted by [Formula: see text]. In this paper, we characterize trees [Formula: see text] of order [Formula: see text] [Formula: see text] such that [Formula: see text] and also establish a strict upper bound for [Formula: see text] for a tree of even order [Formula: see text] [Formula: see text]. We construct some family of graphs [Formula: see text] with [Formula: see text] and prove some results on [Formula: see text]-partitions of [Formula: see text] when [Formula: see text].

scholarly journals Improved Bounds for Radiok-Chromatic Number of HypercubeQn

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Laxman Saha ◽  
Pratima Panigrahi ◽  
Pawan Kumar
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Graph Coloring ◽  
Upper Bound ◽  
Channel Assignment ◽  
Assignment Problem ◽  
Communication Problem ◽  
Channel Assignment Problem

A number of graph coloring problems have their roots in a communication problem known as the channel assignment problem. The channel assignment problem is the problem of assigning channels (nonnegative integers) to the stations in an optimal way such that interference is avoided as reported by Hale (2005). Radiok-coloring of a graph is a special type of channel assignment problem. Kchikech et al. (2005) have given a lower and an upper bound for radiok-chromatic number of hypercubeQn, and an improvement of their lower bound was obtained by Kola and Panigrahi (2010). In this paper, we further improve Kola et al.'s lower bound as well as Kchikeck et al.'s upper bound. Also, our bounds agree for nearly antipodal number ofQnwhenn≡2(mod 4).

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