scholarly journals On the r-dynamic coloring of some fan graph families

2021 ◽  
Vol 29 (3) ◽  
pp. 151-181
Author(s):  
Raúl M. Falcón ◽  
M. Venkatachalam ◽  
S. Gowri ◽  
G. Nandini
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Graph Invariant ◽  
Total Graph ◽  
Sharp Bounds ◽  
Middle Graph ◽  
Graph Families ◽  
Fan Graph

Abstract In this paper, we determine the r-dynamic chromatic number of the fan graph Fm,n and determine sharp bounds of this graph invariant for four related families of graphs: The middle graph M(Fm,n ), the total graph T (Fm,n ), the central graph C(Fm,n ) and the line graph L(Fm,n ). In addition, we determine the r-dynamic chromatic number of each one of these four families of graphs in case of being m = 1.

On b-chromatic number of theta graph families

2020 ◽  
pp. 643-650
Author(s):  
M. Vinitha ◽  
M. Venkatachalam
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Total Graph ◽  
Middle Graph ◽  
Graph Families

In this paper, we investigate the $b$-chromatic number for the theta graph $\theta(s_1, s_2, \cdots, s_n)$, middle graph of theta graph $M(G)$, total graph of theta graph $T(G)$, line graph of theta graph $L(G)$ and the central graph of theta graph $C(G)$.

scholarly journals ON DYNAMIC COLORING OF WEB GRAPH

2018 ◽  
Vol 5 (2) ◽  
pp. 11-15
Author(s):  
Aaresh R.R ◽  
Venkatachalam M ◽  
Deepa T
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Web Graph ◽  
Middle Graph

Dynamic coloring of a graph G is a proper coloring. The chromatic number of a graph G is the minimum k such that G has a dynamic coloring with k colors. In this paper we investigate the dynamic chromatic number for the Central graph, Middle graph, Total graph and Line graph of Web graph Wn denoted by C(Wn), M(Wn), T(Wn) and L(Wn) respectively.

scholarly journals On r-dynamic coloring of comb graphs

2021 ◽  
Vol 27 (2) ◽  
pp. 191-200
Author(s):  
K. Kalaiselvi ◽  
◽  
N. Mohanapriya ◽  
J. Vernold Vivin ◽  
◽  
...  
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Middle Graph ◽  
Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

scholarly journals Omega Index of Line and Total Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Musa Demirci ◽  
Sadik Delen ◽  
Ahmet Sinan Cevik ◽  
Ismail Naci Cangul
Keyword(s):  
Line Graph ◽  
Graph Invariant ◽  
Original Graph ◽  
Total Graph ◽  
Graph Classes ◽  
Number Of Faces

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.

scholarly journals A note on achromatic coloring of star graph families

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 251-255 ◽  
Author(s):  
Vivin Vernold ◽  
M. Venkatachalam ◽  
Ali Akbar
Keyword(s):  
Star Graph ◽  
Total Graph ◽  
Middle Graph ◽  
Graph Families

In this paper, we find the achromatic number of central graph, middle graph and total graph of star graph, denoted by C(K1,n), M(K1,n) and T(K1,n) respectively.

A criterion for the planarity of the total graph of a graph

1967 ◽  
Vol 63 (3) ◽  
pp. 679-681 ◽  
Author(s):  
Mehdi Behzad
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Line Graphs ◽  
Total Graph ◽  
Planar Line ◽  
Total Chromatic Number

Two well-known numbers associated with a graph G (finite and undirected with no loops or multiple lines) are the (point) chromatic and the line chromatic number of G (see (2)). With G there is associated a graph L(G), called the line-graph of G, such that the line chromatic number of G is the same as the chromatic number of L(G). This concept was originated by Whitney (9) in 1932. In 1963, Sedlâček (8) characterized graphs with planar line-graphs. In this note we introduce the notions of the total chromatic number and the total graph of a graph, and characterize graphs with planar total graphs.

scholarly journals Harmonious coloring on double star graph families

2012 ◽  
Vol 43 (2) ◽  
pp. 153-158 ◽  
Author(s):  
Vernold Vivin.J ◽  
Venkatachalam M. ◽  
Kaliraj K.
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Double Star ◽  
Star Graph ◽  
Graph Families

In this present paper, we have proved for the line graph of double star graph, the harmonious chromatic number and the achromatic number are equal. As a motivation this work can be extended by classifying the different families of graphs for which these two numbers are equal.

scholarly journals Sharp Bounds for the General Sum-Connectivity Indices of Transformation Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Haiying Wang ◽  
Jia-Bao Liu ◽  
Shaohui Wang ◽  
Wei Gao ◽  
Shehnaz Akhter ◽  
...  
Keyword(s):  
Real Number ◽  
Line Graph ◽  
Connectivity Index ◽  
Point Graph ◽  
Total Graph ◽  
Sharp Bounds ◽  
Graph Transformations ◽  
Degree Of Vertex ◽  
Connectivity Indices ◽  
Distinct Transformation

Given a graph G, the general sum-connectivity index is defined as χα(G)=∑uv∈E(G)dGu+dGvα, where dG(u) (or dG(v)) denotes the degree of vertex u (or v) in the graph G and α is a real number. In this paper, we obtain the sharp bounds for general sum-connectivity indices of several graph transformations, including the semitotal-point graph, semitotal-line graph, total graph, and eight distinct transformation graphs Guvw, where u,v,w∈+,-.

A modified anti-magic graph labeling for middle graph and total graph related to path

2016 ◽  
Author(s):  
Zareen Tasneem ◽  
Farissa Tafannum ◽  
Maksuda Rahman Anti ◽  
Wali Mohammad Abdullah ◽  
Md. Mahbubur Rahman
Keyword(s):  
Graph Labeling ◽  
Total Graph ◽  
Middle Graph

scholarly journals A Strengthening of Brooks' Theorem for Line Graphs

10.37236/632 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Clique Number ◽  
Line Graphs ◽  
Delta G

We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.

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