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 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.

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 Total Coloring Conjecture for Certain Classes of Graphs

Algorithms ◽  
2018 ◽  
Vol 11 (10) ◽  
pp. 161 ◽  
Author(s):  
R. Vignesh ◽  
J. Geetha ◽  
K. Somasundaram
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Product Line ◽  
Total Coloring ◽  
Lexicographic Product ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Total Coloring Conjecture

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

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].

Total coloring of quasi-line graphs and inflated graphs

2020 ◽  
pp. 2150060
Author(s):  
S. Mohan ◽  
J. Geetha ◽  
K. Somasundaram
Keyword(s):  
Chromatic Number ◽  
Total Coloring ◽  
Line Graphs ◽  
Tight Bound ◽  
Free Graph ◽  
Induced Subgraph ◽  
Total Chromatic Number

A total coloring of a graph is an assignment of colors to all the elements (vertices and edges) of the graph such that no two adjacent or incident elements receive the same color. A claw-free graph is a graph that does not have [Formula: see text] as an induced subgraph. Quasi-line and inflated graphs are two well-known classes of claw-free graphs. In this paper, we prove that the quasi-line and inflated graphs are totally colorable. In particular, we prove the tight bound of the total chromatic number of some classes of quasi-line graphs and inflated graphs.

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 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.

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