scholarly journals Packing chromatic numbers of finite super subdivisions of graphs

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3275-3286
Author(s):  
Rachid Lemdani ◽  
Moncef Abbas ◽  
Jasmina Ferme
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
General Properties ◽  
Vertex Set ◽  
Packing Chromatic Number ◽  
Corona Graphs

Given a graph G and a positive integer i, an i-packing in G is a subset W of the vertex set of G such that the distance between any two distinct vertices from W is greater than i. The packing chromatic number of a graph G, ??(G), is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ? {1,..., k}, where each Vi is an i-packing. In this paper, we present some general properties of packing chromatic numbers of finite super subdivisions of graphs. We determine the packing chromatic numbers of the finite super subdivisions of complete graphs, cycles and some neighborhood corona graphs.

scholarly journals BILANGAN KROMATIK LOKASI DARI GRAF P m P n ; K m P n ; DAN K , m K n

2013 ◽  
Vol 2 (1) ◽  
pp. 14
Author(s):  
Mariza Wenni
Keyword(s):  
Natural Number ◽  
Chromatic Number ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Color Code ◽  
Connected Graphs ◽  
Vertex Set ◽  
Distinct Color

Let G and H be two connected graphs. Let c be a vertex k-coloring of aconnected graph G and let = fCg be a partition of V (G) into the resultingcolor classes. For each v 2 V (G), the color code of v is dened to be k-vector: c1; C2; :::; Ck(v) =(d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; Ci) = minfd(v; x) j x 2 Cg, 1 i k. Ifdistinct vertices have distinct color codes with respect to , then c is called a locatingcoloring of G. The locating chromatic number of G is the smallest natural number ksuch that there are locating coloring with k colors in G. The Cartesian product of graphG and H is a graph with vertex set V (G) V (H), where two vertices (a; b) and (a)are adjacent whenever a = a0and bb02 E(H), or aa0i2 E(G) and b = b, denotedby GH. In this paper, we will study about the locating chromatic numbers of thecartesian product of two paths, the cartesian product of paths and complete graphs, andthe cartesian product of two complete graphs.

scholarly journals The Packing Chromatic Number of Different Jump Sizes of Circulant Graphs

2021 ◽  
Vol 33 (5) ◽  
pp. 66-73
Author(s):  
B. CHALUVARAJU ◽  
◽  
M. KUMARA ◽  
Keyword(s):  
Chromatic Number ◽  
Pairwise Distance ◽  
Circulant Graphs ◽  
Vertex Set ◽  
Packing Chromatic Number ◽  
Jump Sizes

The packing chromatic number χ_{p}(G) of a graph G = (V,E) is the smallest integer k such that the vertex set V(G) can be partitioned into disjoint classes V1 ,V2 ,...,Vk , where vertices in Vi have pairwise distance greater than i. In this paper, we compute the packing chromatic number of circulant graphs with different jump sizes._{}

On the b-Chromatic Sum of Mycielskian of Km, n, Kn and Cn

2020 ◽  
Vol 20 (02) ◽  
pp. 2050007
Author(s):  
P. C. LISNA ◽  
M. S. SUNITHA
Keyword(s):  
Image Processing ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Proper Coloring ◽  
Color Class ◽  
Vertex Set ◽  
Chromatic Sum ◽  
Complete Bipartite Graphs

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 φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v1, v2,…, vn} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u1, u2, …, un} joining u to each vertex ui(1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

scholarly journals Packing chromatic number of transformation graphs

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 1991-1995
Author(s):  
Derya Durgun ◽  
Busra Ozen-Dortok
Keyword(s):  
Chromatic Number ◽  
Graph Coloring ◽  
Chromatic Numbers ◽  
Np Complete ◽  
Packing Chromatic Number ◽  
General Graphs

Graph coloring is an assignment of labels called colors to elements of a graph. The packing coloring was introduced by Goddard et al. [1] in 2008 which is a kind of coloring of a graph. This problem is NP-complete for general graphs. In this paper, we consider some transformation graphs and generalized their packing chromatic numbers.

scholarly journals The conjugacy class graphs of non-abelian 3-groups

2020 ◽  
Vol 16 (3) ◽  
pp. 297-299
Author(s):  
Athirah Zulkarnain ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Conjugacy Class ◽  
Chromatic Number ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
Prime Power Order ◽  
Complete Graphs ◽  
Dominating Number ◽  
Minimum Number ◽  
Vertex Set ◽  
Class Graph

A graph is formed by a pair of vertices and edges. It can be related to groups by using the groups’ properties for its vertices and edges. The set of vertices of the graph comprises the elements or sets from the group while the set of edges of the graph is the properties and condition for the graph. A conjugacy class of an element  is the set of elements that are conjugated with . Any element of a group , labelled as , is conjugated to  if it satisfies  for some elements  in  with its inverse . A conjugacy class graph of a group   is defined when its vertex set is the set of non-central conjugacy classes of  . Two distinct vertices   and   are connected by an edge if and only if their cardinalities are not co-prime, which means that the order of the conjugacy classes of  and  have common factors. Meanwhile, a simple graph is the graph that contains no loop and no multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is adjacent. Moreover, a  -group is the group with prime power order. In this paper, the conjugacy class graphs for some non-abelian 3-groups are determined by using the group’s presentations and the definition of conjugacy class graph. There are two classifications of the non-abelian 3-groups which are used in this research. In addition, some properties of the conjugacy class graph such as the chromatic number, the dominating number, and the diameter are computed. A chromatic number is the minimum number of vertices that have the same colours where the adjacent vertices have distinct colours. Besides, a dominating number is the minimum number of vertices that is required to connect all the vertices while a diameter is the longest path between any two vertices. As a result of this research, the conjugacy class graphs of these groups are found to be complete graphs with chromatic number, dominating number and diameter that are equal to eight, one and one, respectively.

scholarly journals On star coloring of Mycielskians

2018 ◽  
Vol 2 (2) ◽  
pp. 82
Author(s):  
K. Kaliraj ◽  
V. Kowsalya ◽  
Vernold Vivin
Keyword(s):  
Chromatic Number ◽  
Graph Transformation ◽  
Clique Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Complete Bipartite Graphs ◽  
Star Coloring

<p>In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph <span class="math"><em>G</em></span> into a new graph <span class="math"><em>μ</em>(<em>G</em>)</span>, we now call the Mycielskian of <span class="math"><em>G</em></span>, which has the same clique number as <span class="math"><em>G</em></span> and whose chromatic number equals <span class="math"><em>χ</em>(<em>G</em>) + 1</span>. In this paper, we find the star chromatic number for the Mycielskian graph of complete graphs, paths, cycles and complete bipartite graphs.</p>

Disjointness graphs of segments in the space

2020 ◽  
pp. 1-15
Author(s):  
János Pach ◽  
Gábor Tardos ◽  
Géza Tóth
Keyword(s):  
Chromatic Number ◽  
Pairwise Disjoint ◽  
Clique Number ◽  
Upper Bounds ◽  
Efficient Algorithms ◽  
Np Hard ◽  
Chromatic Numbers ◽  
Lines In Space ◽  
Vertex Set

Abstract The disjointness graph G = G(𝒮) of a set of segments 𝒮 in ${\mathbb{R}^d}$ , $$d \ge 2$$ , is a graph whose vertex set is 𝒮 and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies $\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$ , where ω(G) denotes the clique number of G. It follows that 𝒮 has Ω(n1/5) pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing ω(G) and χ(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colourings of G in which the number of colours satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (ω(G) = 2), but whose chromatic numbers are arbitrarily large.

Radio k-chromatic number of cycles for large k

2017 ◽  
Vol 09 (03) ◽  
pp. 1750031 ◽  
Author(s):  
Nathaniel Karst ◽  
Joshua Langowitz ◽  
Jessica Oehrlein ◽  
Denise Sakai Troxell
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Lower Bounds ◽  
Vertex Set ◽  
Number Of Cycles

For a positive integer [Formula: see text], a radio k-labeling of a graph [Formula: see text] is a function [Formula: see text] from its vertex set to the non-negative integers such that for all pairs of distinct vertices [Formula: see text] and [Formula: see text], we have [Formula: see text] where [Formula: see text] is the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The minimum span over all radio [Formula: see text]-labelings of [Formula: see text] is called the radio k-chromatic number and denoted by [Formula: see text]. The most extensively studied cases are the classic vertex colorings ([Formula: see text]), [Formula: see text](2,1)-labelings ([Formula: see text]), radio labelings ([Formula: see text], the diameter of [Formula: see text]), and radio antipodal labelings ([Formula: see text]. Determining exact values or tight bounds for [Formula: see text] is often non-trivial even within simple families of graphs. We provide general lower bounds for [Formula: see text] for all cycles [Formula: see text] when [Formula: see text] and show that these bounds are exact values when [Formula: see text].

Coloring 3-power of 3-subdivision of subcubic graph

2018 ◽  
Vol 10 (03) ◽  
pp. 1850041 ◽  
Author(s):  
Fang Wang ◽  
Xiaoping Liu
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
London Math ◽  
Total Coloring ◽  
Total Graph ◽  
Total Chromatic Number ◽  
Subcubic Graph ◽  
Minimum Integer ◽  
Vertex Set ◽  
Number Formula

Let [Formula: see text] be a graph and [Formula: see text] be a positive integer. The [Formula: see text]-subdivision [Formula: see text] of [Formula: see text] is the graph obtained from [Formula: see text] by replacing each edge by a path of length [Formula: see text]. The [Formula: see text]-power [Formula: see text] of [Formula: see text] is the graph with vertex set [Formula: see text] in which two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if the distance [Formula: see text] between [Formula: see text] and [Formula: see text] in [Formula: see text] is at most [Formula: see text]. Note that [Formula: see text] is the total graph [Formula: see text] of [Formula: see text]. The chromatic number [Formula: see text] of [Formula: see text] is the minimum integer [Formula: see text] for which [Formula: see text] has a proper [Formula: see text]-coloring. The total chromatic number of [Formula: see text], denoted by [Formula: see text], is the chromatic number of [Formula: see text]. Rosenfeld [On the total coloring of certain graphs, Israel J. Math. 9 (1971) 396–402] and independently, Vijayaditya [On total chromatic number of a graph, J. London Math. Soc. 2 (1971) 405–408] showed that for a subcubic graph [Formula: see text], [Formula: see text]. In this note, we prove that for a subcubic graph [Formula: see text], [Formula: see text].

scholarly journals A Note on Packing Chromatic Number of the Square Lattice

10.37236/466 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Roman Soukal ◽  
Přemysl Holub
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Assignment Problem ◽  
Pairwise Distance ◽  
Frequency Assignment ◽  
Square Lattice ◽  
Frequency Assignment Problem ◽  
Vertex Set ◽  
Packing Chromatic Number

The concept of a packing colouring is related to a frequency assignment problem. The packing chromatic number $\chi_p(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set $V (G)$ can be partitioned into disjoint classes $X_1, \dots, X_k$, where vertices in $X_i$ have pairwise distance greater than $i$. In this note we improve the upper bound on the packing chromatic number of the square lattice.

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