scholarly journals Game Chromatic Number of Generalized Petersen Graphs and Jahangir Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Ramy Shaheen ◽  
Ziad Kanaya ◽  
Khaled Alshehada
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Game Chromatic Number ◽  
Generalized Petersen Graphs ◽  
Minimum Number ◽  
Petersen Graphs ◽  
Jahangir Graph ◽  
Prism Graph

Let G = V , E be a graph, and two players Alice and Bob alternate turns coloring the vertices of the graph G a proper coloring where no two adjacent vertices are signed with the same color. Alice's goal is to color the set of vertices using the minimum number of colors, which is called game chromatic number and is denoted by χ g G , while Bob's goal is to prevent Alice's goal. In this paper, we investigate the game chromatic number χ g G of Generalized Petersen Graphs G P n , k for k ≥ 3 and arbitrary n , n -Crossed Prism Graph, and Jahangir Graph J n , m .

On strict strong coloring of graphs

2020 ◽  
pp. 2150040
Author(s):  
A. Mohammed Abid ◽  
T. R. Ramesh Rao
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Color Class ◽  
Minimum Number ◽  
Strong Chromatic Number

A strict strong coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which every vertex of the graph is adjacent to every vertex of some color class. The minimum number of colors required for a strict strong coloring of [Formula: see text] is called the strict strong chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we characterize the results on strict strong coloring of Mycielskian graphs and iterated Mycielskian graphs.

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.

CERTAIN OPERATION OF GENERALIZED PETERSEN GRAPHS HAVING LOCATING-CHROMATIC NUMBER FIVE

2020 ◽  
Vol 24 (2) ◽  
pp. 83-97
Author(s):  
Agus Irawan ◽  
Asmiati ◽  
S. Suharsono ◽  
Kurnia Muludi ◽  
La Zakaria
Keyword(s):  
Chromatic Number ◽  
Generalized Petersen Graphs ◽  
Petersen Graphs

On the dominator coloring in proper interval graphs and block graphs

2015 ◽  
Vol 07 (04) ◽  
pp. 1550043 ◽  
Author(s):  
B. S. Panda ◽  
Arti Pandey
Keyword(s):  
Chromatic Number ◽  
Dominating Set ◽  
Domination Number ◽  
Maximum Size ◽  
Interval Graphs ◽  
Proper Coloring ◽  
Minimum Cardinality ◽  
Block Graphs ◽  
Minimum Number ◽  
Dominator Coloring

In a graph [Formula: see text], a vertex [Formula: see text] dominates a vertex [Formula: see text] if either [Formula: see text] or [Formula: see text] is adjacent to [Formula: see text]. A subset of vertex set [Formula: see text] that dominates all the vertices of [Formula: see text] is called a dominating set of graph [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text] and is denoted by [Formula: see text]. A proper coloring of a graph [Formula: see text] is an assignment of colors to the vertices of [Formula: see text] such that any two adjacent vertices get different colors. The minimum number of colors required for a proper coloring of [Formula: see text] is called the chromatic number of [Formula: see text] and is denoted by [Formula: see text]. A dominator coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] such that every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of [Formula: see text] is called the dominator chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we study the dominator chromatic number for the proper interval graphs and block graphs. We show that every proper interval graph [Formula: see text] satisfies [Formula: see text], and these bounds are sharp. For a block graph [Formula: see text], where one of the end block is of maximum size, we show that [Formula: see text]. We also characterize the block graphs with an end block of maximum size and attaining the lower bound.

scholarly journals Families of graphs with maximum nullity equal to zero forcing number

2018 ◽  
Vol 6 (1) ◽  
pp. 56-67
Author(s):  
Joseph S. Alameda ◽  
Emelie Curl ◽  
Armando Grez ◽  
Leslie Hogben ◽  
O’Neill Kingston ◽  
...  
Keyword(s):  
Color Change ◽  
Simple Graph ◽  
Zero Forcing ◽  
Generalized Petersen Graphs ◽  
Zero Forcing Number ◽  
Regularity Properties ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Minimum Number ◽  
Petersen Graphs

Abstract The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i =6 j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartite 3- semiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to find bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen graphs.

scholarly journals More Counterexamples to the Alon-Saks-Seymour and Rank-Coloring Conjectures

10.37236/513 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Sebastian M. Cioabă ◽  
Michael Tait
Keyword(s):  
Computational Complexity ◽  
Chromatic Number ◽  
Adjacency Matrix ◽  
Proper Coloring ◽  
Partition Number ◽  
Similar Work ◽  
Minimum Number

The chromatic number $\chi(G)$ of a graph $G$ is the minimum number of colors in a proper coloring of the vertices of $G$. The biclique partition number ${\rm bp}(G)$ is the minimum number of complete bipartite subgraphs whose edges partition the edge-set of $G$. The Rank-Coloring Conjecture (formulated by van Nuffelen in 1976) states that $\chi(G)\leq {\rm rank}(A(G))$, where ${\rm rank}(A(G))$ is the rank of the adjacency matrix of $G$. This was disproved in 1989 by Alon and Seymour. In 1991, Alon, Saks, and Seymour conjectured that $\chi(G)\leq {\rm bp}(G)+1$ for any graph $G$. This was recently disproved by Huang and Sudakov. These conjectures are also related to interesting problems in computational complexity. In this paper, we construct new infinite families of counterexamples to both the Alon-Saks-Seymour Conjecture and the Rank-Coloring Conjecture. Our construction is a generalization of similar work by Razborov, and Huang and Sudakov.

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 Power Dominator Chromatic Number for some Special Graphs

2019 ◽  
Vol 8 (12) ◽  
pp. 3957-3960
Keyword(s):  
Chromatic Number ◽  
Star Graph ◽  
Proper Coloring ◽  
Induction Method ◽  
Color Class ◽  
Wheel Graph ◽  
Minimum Number ◽  
Fan Graph ◽  
Multiple Edges ◽  
Dominator Coloring

Let G = (V, E) be a finite, connected, undirected with no loops, multiple edges graph. Then the power dominator coloring of G is a proper coloring of G, such that each vertex of G power dominates every vertex of some color class. The minimum number of color classes in a power dominator coloring of the graph, is the power dominator chromatic number . Here we study the power dominator chromatic number for some special graphs such as Bull Graph, Star Graph, Wheel Graph, Helm graph with the help of induction method and Fan Graph. Suitable examples are provided to exemplify the results.

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