scholarly journals Geodetic Number and Geo-Chromatic Number of 2-Cartesian Product of Some Graphs

2022 ◽  
Vol 12 (01) ◽  
pp. 1-16
Author(s):  
Medha Itagi Huilgol ◽  
B. Divya
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Geodetic Number

scholarly journals Chromatic Number for a Generalization of Cartesian Product Graphs

10.37236/160 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Daniel Král' ◽  
Douglas B. West
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Rectangular Grid ◽  
Product Graphs ◽  
Cartesian Product Graphs

Let ${\cal G}$ be a class of graphs. A $d$-fold grid over ${\cal G}$ is a graph obtained from a $d$-dimensional rectangular grid of vertices by placing a graph from ${\cal G}$ on each of the lines parallel to one of the axes. Thus each vertex belongs to $d$ of these subgraphs. The class of $d$-fold grids over ${\cal G}$ is denoted by ${\cal G}^d$. Let $f({\cal G};d)=\max_{G\in{\cal G}^d}\chi(G)$. If each graph in ${\cal G}$ is $k$-colorable, then $f({\cal G};d)\le k^d$. We show that this bound is best possible by proving that $f({\cal G};d)=k^d$ when ${\cal G}$ is the class of all $k$-colorable graphs. We also show that $f({\cal G};d)\ge{\left\lfloor\sqrt{{d\over 6\log d}}\right\rfloor}$ when ${\cal G}$ is the class of graphs with at most one edge, and $f({\cal G};d)\ge {\left\lfloor{d\over 6\log d}\right\rfloor}$ when ${\cal G}$ is the class of graphs with maximum degree $1$.

The b -chromatic number of the cartesian product of two graphs

2007 ◽  
Vol 44 (1) ◽  
pp. 49-55
Author(s):  
Mekkia Kouider ◽  
Maryvonne Mahéo
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Proper Coloring

In this paper we study the b -chromatic number of the cartesian product of two graphs. The b -chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G , such that we obtain a proper coloring and each color i has at least one representative χi adjacent to a vertex of every color j , 1 ≦ j ≠ i ≦ k . In this paper we get ρ( G□H ) ≦ ρ( G )( nH + 1) + Δ( H ) + 1, when the girth of G is assumed to be greater than or equal to 7.

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.

Star chromatic number of some graphs

2021 ◽  
pp. 2150089
Author(s):  
S. Akbari ◽  
M. CHAVOOSHI ◽  
M. Ghanbari ◽  
S. Taghian
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Vertex Coloring ◽  
Star Coloring ◽  
Proper Vertex

A proper vertex coloring of a graph [Formula: see text] is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored [Formula: see text] in [Formula: see text]. In this paper, we show that the Cartesian product of any two cycles, except [Formula: see text] and [Formula: see text], has a [Formula: see text]-star coloring.

scholarly journals k -tuple chromatic number of the cartesian product of graphs

2015 ◽  
Vol 50 ◽  
pp. 243-248
Author(s):  
Flavia Bonomo ◽  
Ivo Koch ◽  
Pablo Torres ◽  
Mario Valencia-Pabon
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

On construction of fuzzy chromatic number of cartesian product of path and other fuzzy graphs

2020 ◽  
Vol 39 (1) ◽  
pp. 1073-1080
Author(s):  
Isnaini Rosyida ◽  
Widodo ◽  
Ch. Rini Indrati ◽  
Diari Indriati
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Fuzzy Graphs

scholarly journals Strong Geodetic Number in Some Networks

2019 ◽  
Vol 11 (2) ◽  
pp. 20
Author(s):  
Huifen Ge ◽  
Zhao Wang ◽  
Jinyu Zou
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Geodetic Number ◽  
Product Graphs ◽  
Geodetic Set ◽  
Cartesian Product Graphs ◽  
Strong Geodetic Number

A vertex subset S of a graph is called a strong geodetic set if there exists a choice of exactly one geodesic for each pair of vertices of S in such a way that these (|S| 2) geodesics cover all the vertices of graph G. The strong geodetic number of G, denoted by sg(G), is the smallest cardinality of a strong geodetic set. In this paper, we give an upper bound of strong geodetic number of the Cartesian product graphs and study this parameter for some Cartesian product networks.

scholarly journals A Comparison on the Bounds of Chromatic Preserving Number and Dom-Chromatic Number of Cartesian Product and Kronecker Product of Paths

2012 ◽  
Vol 11 (4) ◽  
pp. 43-58
Author(s):  
T N Janakiraman ◽  
M Poobalaranjani
Keyword(s):  
Comparative Study ◽  
Chromatic Number ◽  
Kronecker Product ◽  
Dominating Set ◽  
Cartesian Product ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Vertex Set

Let G be a simple graph with vertex set V and edge set E. A Set S Í V is said to be a chromatic preserving set or a cp-set if χ(<S>) = χ(G) and the minimum cardinality of a cp-set in G is called the chromatic preserving number or cp-number of G and is denoted by cpn(G). A cp-set of cardinality cpn(G) is called a cpn-set. A subset S of V is said to be a dom- chromatic set (or a dc-set) if S is a dominating set and χ(<S>) = χ(G). The minimum cardinality of a dom-chromatic set in a graph G is called the dom-chromatic number (or dc- number) of G and is denoted by γch(G). The Kronecker product G1 Ù G2 of two graphs G1 = (V1, E1) and G2 = (V2, E2) is the graph G with vertex set V1 x V2 and any two distinct vertices (u1, v1) and (u2, v2) of G are adjacent if u1u2 Î E1 and v1v2 Î E2. The Cartesian product G1 x G2 is the graph with vertex set V1 x V2 where any two distinct vertices (u1, v1) and (u2, v2) are adjacent whenever (i) u1 = u2 and v1v2 Î E2 or (ii) u1u2 Î E1 and v1 = v2. These two products have no common edges. They are almost like complements but not exactly. In this paper, we discuss the behavior of the cp-number and dc-number and their bounds for product of paths in the two cases. A detailed comparative study is also done.

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