scholarly journals The Locating Chromatic Number of Book Graph

2021 ◽  
Vol 2021 ◽  
pp. 1-3
Author(s):  
Nur Inayah ◽  
Wisnu Aribowo ◽  
Maiyudi Mariska Windra Yahya
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Color Code ◽  
Distinct Color

Let G = V G , E G be a connected graph and c : V G ⟶ 1,2 , … , k be a proper k -coloring of G . Let Π be a partition of vertices of G induced by the coloring c . We define the color code c Π v of a vertex v ∈ V G as an ordered k -tuple that contains the distance between each partition to the vertex v . If distinct vertices have distinct color code, then c is called a locating k -coloring of G . The locating chromatic number of G is the smallest k such that G has a locating k -coloring. In this paper, we determine the locating chromatic number of book graph.

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 Locating Chromatic Number of Powers of Paths and Cycles

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 389
Author(s):  
Manal Ghanem ◽  
Hasan Al-Ezeh ◽  
Ala’a Dabbour
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Minimum Distance ◽  
Upper And Lower Bounds ◽  
Color Code ◽  
Partition Class ◽  
Distinct Color

Let c be a proper k-coloring of a graph G. Let π = { R 1 , R 2 , … , R k } be the partition of V ( G ) induced by c, where R i is the partition class receiving color i. The color code c π ( v ) of a vertex v of G is the ordered k-tuple ( d ( v , R 1 ) , d ( v , R 2 ) , … , d ( v , R k ) ) , where d ( v , R i ) is the minimum distance from v to each other vertex u ∈ R i for 1 ≤ i ≤ k . If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by χ L ( G ) , is the smallest k such that G admits a locating coloring with k colors. In this paper, we give a characterization of the locating chromatic number of powers of paths. In addition, we find sharp upper and lower bounds for the locating chromatic number of powers of cycles.

scholarly journals BILANGAN KROMATIK LOKASI UNTUK GRAF AMALGAMASI BINTANG

2013 ◽  
Vol 2 (1) ◽  
pp. 6
Author(s):  
Fadhilah Syamsi
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Color Code ◽  
Coloring Number ◽  
Number Of Leaves ◽  
Distinct Color

Let G = (V; E) be a connected graph and c a coloring of G. For i = 1; 2; :::; k,we dene the color classes Cias the set of vertices receiving color i. The color code c(v)of a vertex v 2 V (G) is the k-vector (d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; C) isthe distance between v and C. If all vertices of G have distinct color codes, then c iscalled a locating-coloring of G. The locating-coloring number of graph G, denoted byLi(G), is the smallest positive integer k such that G has a locating coloring with k color.Let K1;nibe star, where niis the number of leaves of each star K. We dene thevertex amalgamation of star, denoted by Sk;(n1;:::;nk)1;n, as a graph obtained from starsKby identifying one arbitrary leaf from each star. We dene the edge amalgamationof star, denoted by S1;nik;(n, as a graph obtained by uniting an edge of each star.If ni1;:::;nk)= m for each i, then we denoted the vertex amalgamation of star as Sandthe edge amalgamation of star as Sk;m. In this paper we discuss the locating coloring ofSk;(n1;:::;nk)and Sk;(n1;:::;nk).

scholarly journals Further Results on Locating Chromatic Number for Amalgamation of Stars Linking by One Path

2018 ◽  
Vol 2 (1) ◽  
pp. 50
Author(s):  
A. Asmiati ◽  
Lyra Yulianti ◽  
C. Ike Tri Widyastuti
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Proper Coloring ◽  
Color Code ◽  
Color Class ◽  
Different Color

Let G = (V,E) be a connected graph. Let c be a proper coloring using k colors, namely 1, 2,·s, k. Let <span style="font-family: symbol;">P</span>={S<sub>1</sub>, S<sub>2</sub>,..., S<sub>k</sub>} be a partition of V(G) induced by c and let S<sub>i</sub> be the color class that receives the color i. The color code, c<sub><span style="font-family: symbol;">P</span></sub>(v)=(d(v,S<sub>1</sub>), d(v,S<sub>2</sub>),...,d(v,S<sub>k</sub>)), where d(v,S<sub>i</sub>)=min {d(v,x)|x <span style="font-family: symbol;">Î</span> S<sub>i</sub>} for i <span style="font-family: symbol;">Î</span> [1,k]. If all vertices in V(G) have different color codes, then c is called as the \emphlocating-chromatic k-coloring of G. Minimum k such that G has the locating-chromatic k-coloring is called the locating-chromatic number, denoted by <span style="font-family: symbol;">c</span><sub>L</sub>(G). In this paper, we discuss the locating-chromatic number for n certain amalgamation of stars linking a path, denoted by nS<sub>k,m</sub>, for n ≥ 1, m ≥ 2, k ≥ 3, and k&gt;m.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

scholarly journals Large Induced Subgraphs with All Degrees Odd

1992 ◽  
Vol 1 (4) ◽  
pp. 335-349 ◽  
Author(s):  
A. D. Scott
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Connected Graph ◽  
Analogous Problem ◽  
Induced Subgraphs ◽  
Induced Subgraph

We prove that every connected graph of order n ≥ 2 has an induced subgraph with all degrees odd of order at least cn/log n, where cis a constant. We also give a bound in terms of chromatic number, and resolve the analogous problem for random graphs.

CO-MAXIMAL IDEAL GRAPHS OF COMMUTATIVE RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250114 ◽  
Author(s):  
MENG YE ◽  
TONGSUO WU
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Maximal Ideal ◽  
Connected Graph ◽  
Jacobson Radical ◽  
Clique Number ◽  
Commutative Rings ◽  
Maximal Ideals ◽  
Simple Connected Graph

In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use [Formula: see text] to denote this graph, with its vertices the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. We show some properties of this graph. For example, this graph is a simple, connected graph with diameter less than or equal to three, and both the clique number and the chromatic number of the graph are equal to the number of maximal ideals of the ring R.

THE RAMSEY NUMBER FOR A LINEAR FOREST VERSUS TWO IDENTICAL COPIES OF COMPLETE GRAPHS

2010 ◽  
Vol 02 (04) ◽  
pp. 437-444 ◽  
Author(s):  
I. WAYAN SUDARSANA ◽  
HILDA ASSIYATUN ◽  
ADIWIJAYA ◽  
SELVY MUSDALIFAH
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Ramsey Number ◽  
Complete Graphs ◽  
Cocktail Party ◽  
Linear Forest ◽  
Good For

Let H be a graph with the chromatic number h and the chromatic surplus s. A connected graph G of order n is called H-good if R(G, H) = (n - 1)(h - 1) + s. In this paper, we show that Pn is 2Km-good for n ≥ 3. Furthermore, we obtain the Ramsey number R(L, 2Km), where L is a linear forest. Moreover, we also give the Ramsey number R(L, Hm) which is an extension for R(kPn, Hm) proposed by Ali et al. [1], where Hm is a cocktail party graph on 2m vertices.

EQUITABLE COLORING OF 2-DEGENERATE GRAPH AND PLANE GRAPHS WITHOUT CYCLES OF SPECIFIC LENGTHS

2010 ◽  
Vol 02 (02) ◽  
pp. 207-211 ◽  
Author(s):  
YUEHUA BU ◽  
QIONG LI ◽  
SHUIMING ZHANG
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Connected Graph ◽  
Plane Graphs ◽  
Equitable Coloring ◽  
Odd Cycle ◽  
Equitable Chromatic Number

The equitable chromatic number χe(G) of a graph G is the smallest integer k for which G has a proper k-coloring such that the number of vertices in any two color classes differ by at most one. In 1973, Meyer conjectured that the equitable chromatic number of a connected graph G, which is neither a complete graph nor an odd cycle, is at most Δ(G). We prove that this conjecture holds for 2-degenerate graphs with Δ(G) ≥ 5 and plane graphs without 3, 4 and 5 cycles.

scholarly journals On resolving edge colorings in graphs

2003 ◽  
Vol 2003 (46) ◽  
pp. 2947-2959
Author(s):  
Varaporn Saenpholphat ◽  
Ping Zhang
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Edge Colorings

We study the relationships between the resolving edge chromatic number and other graphical parameters and provide bounds for the resolving edge chromatic number of a connected graph.

Sign in / Sign up

Export Citation Format

Share Document