scholarly journals On Neighbor chromatic number of grid and torus graphs

2019 ◽  
Vol 27 (1) ◽  
pp. 3-15
Author(s):  
B. Chaluvaraju ◽  
C. Appajigowda
Keyword(s):  
Chromatic Number ◽  
Grid Graph ◽  
Colored Graph ◽  
Cartesian Products ◽  
Minimum Cardinality ◽  
Color Class

Abstract A set S ⊆ V is a neighborhood set of a graph G = (V, E), if G = ∪v∈S 〈 N[v] 〉, where 〈 N[v] 〉 is the subgraph of a graph G induced by v and all vertices adjacent to v. A neighborhood set S is said to be a neighbor coloring set if it contains at least one vertex from each color class of a graph G, where color class of a colored graph is the set of vertices having one particular color. The neighbor chromatic number χn (G) is the minimum cardinality of a neighbor coloring set of a graph G. In this article, some results on neighbor chromatic number of Cartesian products of two paths (grid graph) and cycles (torus graphs) are explored.

scholarly journals Prime Coloring of Crossing Number Zero Graphs

2019 ◽  
Vol 8 (1) ◽  
pp. 15-19
Author(s):  
P. Murugarajan ◽  
R. Aruldoss
Keyword(s):  
Chromatic Number ◽  
Crossing Number ◽  
Minimum Cardinality ◽  
Color Class ◽  
Class C ◽  
Multiple Edges

In this paper, prime coloring and its chromatic number of some crossing number zero graphs are depicted and its results are vali-dated with few theorems. Prime Coloring is defined as G be a loop less and Without multiple edges with n distinct Vertices on Color class C={c1,c2,c3,…..cn} a bijection ψ:V {c1,c2,c3,…..cn} if for each edge e = cicj ,i≠j , gcd{ ψ (ci), ψ (cj)}=1, ψ (ci) and ψ (cj) receive distinct Colors. The Chromatic number of Prime coloring is minimum cardinality taken by all the Prime colors. It is denoted by η (G).

On the b-chromatic number of cartesian products

2018 ◽  
Vol 239 ◽  
pp. 82-93 ◽  
Author(s):  
Chuan Guo ◽  
Mike Newman
Keyword(s):  
Chromatic Number ◽  
Cartesian Products

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.

Modular coloring of the Cartesian products Km□Kn, Km□Cn, and Km□Pn

2017 ◽  
Vol 09 (06) ◽  
pp. 1750075
Author(s):  
R. Rajarajachozhan ◽  
R. Sampathkumar
Keyword(s):  
Chromatic Number ◽  
Cartesian Products

A modular [Formula: see text]-coloring, [Formula: see text] of a graph [Formula: see text] is a coloring of the vertices of [Formula: see text] with the elements in [Formula: see text] the set of integers modulo [Formula: see text] having the property that for every two adjacent vertices of [Formula: see text] the sums of the colors of their neighbors are different in [Formula: see text] The minimum [Formula: see text] for which [Formula: see text] has a modular [Formula: see text]-coloring is the modular chromatic number of [Formula: see text] This paper is concerned with the modular chromatic number of the Cartesian products [Formula: see text] [Formula: see text] and [Formula: see text]

On the b-chromatic number of some graph products

2012 ◽  
Vol 49 (2) ◽  
pp. 156-169 ◽  
Author(s):  
Marko Jakovac ◽  
Iztok Peterin
Keyword(s):  
Chromatic Number ◽  
Upper And Lower Bounds ◽  
Vertex Coloring ◽  
Graph Products ◽  
Lexicographic Product ◽  
Arbitrary Graph ◽  
Strong Product ◽  
Color Class ◽  
Complete Bipartite Graphs ◽  
Proper Vertex

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a b-coloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G ⊠ H, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of Pn ⊠ H, Cn ⊠ H, Pn[H], Cn[H], and Km,n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.

scholarly journals Introduction to dominated edge chromatic number of a graph

2021 ◽  
Vol 41 (2) ◽  
pp. 245-257
Author(s):  
Mohammad R. Piri ◽  
Saeid Alikhani
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Color Class ◽  
Proper Edge Coloring ◽  
Minimum Number

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph \(G\), is a proper edge coloring of \(G\) such that each color class is dominated by at least one edge of \(G\). The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by \(\chi_{dom}^{\prime}(G)\). We obtain some properties of \(\chi_{dom}^{\prime}(G)\) and compute it for specific graphs. Also examine the effects on \(\chi_{dom}^{\prime}(G)\), when \(G\) is modified by operations on vertex and edge of \(G\). Finally, we consider the \(k\)-subdivision of \(G\) and study the dominated edge chromatic number of these kind of graphs.

scholarly journals Exact 2-Distance b-Coloring and Exact 2-Distance b-Continuity of Helm Graph 𝑯𝒏

YMER Digital ◽  
2021 ◽  
Vol 20 (10) ◽  
pp. 62-72
Author(s):  
S Saraswathi ◽  
◽  
M Poobalaranjani ◽  
Keyword(s):  
Chromatic Number ◽  
Vertex Coloring ◽  
Class A ◽  
Color Class

An exact 2-distance coloring of a graph 𝐺 is a coloring of vertices of 𝐺 such that any two vertices which are at distance exactly 2 receive distinct colors. An exact 2-distance chromatic number𝑒2(𝐺) of 𝐺 is the minimum 𝑘 for which 𝐺 admits an exact 2-distance coloring with 𝑘 colors. A 𝑏-coloring of 𝐺 by 𝑘 colors is a proper 𝑘-vertex coloring such that in each color class, there exists a vertex called a color dominating vertex which has a neighbor in every other color class. A vertex that has a 2-neighbor in all other color classes is called an exact 2-distance color dominating vertex (or an 𝑒2-cdv). Exact 2-distance 𝑏-coloring (or an 𝑒2𝑏-coloring) of 𝐺 is an exact 2-distance coloring such that each color class contains an 𝑒2- cdv. An exact 2-distance 𝑏-chromatic number (or an 𝑒2𝑏-number) 𝑒2𝑏(𝐺) of 𝐺 is the largest integer 𝑘 such that 𝐺 has an 𝑒2𝑏-coloring with 𝑘colors. If for each integer𝑘, 𝑒2(𝐺) ≤ 𝑘 ≤ 𝑒2𝑏(𝐺), 𝐺 has an 𝑒2𝑏-coloring by 𝑘 colors, then 𝐺 is said to be an exact 2-distance 𝑏- continuous graph. In this paper, the 𝑒2𝑏-number𝑒2𝑏(𝐻𝑛)of the helm graph 𝐻𝑛is obtained and 𝑒2𝑏-continuity of 𝐻𝑛is discussed.

scholarly journals 2-Distance Strong b-coloring of Perfect 𝜟-ary Tree

YMER Digital ◽  
2021 ◽  
Vol 20 (10) ◽  
pp. 196-206
Author(s):  
M Poobalaranjani ◽  
◽  
S Saraswathi ◽  
Keyword(s):  
Chromatic Number ◽  
Rooted Tree ◽  
Exact Bound ◽  
Class A ◽  
Color Class

A 2-distance 𝑏-coloring is a 2-distance coloring in which every color class contains a vertex which has a neighbor in every other color class. A 2-distance strong 𝑏-coloring (2𝑠𝑏- coloring) is a 2-distance coloring in which every color class contains a vertex 𝑢 such that there is a vertex 𝑣 in every other color class satisfying the condition that the distance between 𝑢 and 𝑣 is at most 2. The 2-distance 𝑏-chromatic number 𝜒2𝑏(𝐺) (2𝑏-number) is the largest integer 𝑘 such that 𝐺 admits a 2-distance 𝑏-coloring with 𝑘 colors and the 2-distance strong bchromatic number 𝜒2𝑠𝑏(𝐺) (2𝑠𝑏-number) is the maximum k such that 𝐺 admits a 2𝑠𝑏-coloring with 𝑘 colors. A tree with a special vertex called the root is called a rooted tree. A perfect 𝛥- ary tree, is a rooted tree in which all internal vertices are of degree 𝛥 and all pendant vertices are at the same level. In this paper, the exact bound of the 2𝑠𝑏-number of perfect 𝛥-ary tree are obtained.

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.

b-Chromatic sum of a graph

2015 ◽  
Vol 07 (04) ◽  
pp. 1550040 ◽  
Author(s):  
P. C. Lisna ◽  
M. S. Sunitha
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Double Star ◽  
Star Graph ◽  
Proper Coloring ◽  
Color Class ◽  
Wheel Graph ◽  
Chromatic Sum

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 [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.

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