Semi-total graph colourings, the beta parameter, and total chromatic number

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].

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100041657 ◽

1967 ◽

Vol 63 (3) ◽

pp. 679-681 ◽

Cited By ~ 36

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.

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Total chromatic number of one kind of join graphs

Discrete Mathematics ◽

10.1016/j.disc.2006.03.056 ◽

2006 ◽

Vol 306 (16) ◽

pp. 1895-1905 ◽

Cited By ~ 9

Author(s):

Guangrong Li ◽

Limin Zhang

Keyword(s):

Chromatic Number ◽

Total Chromatic Number

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A Characterization of Graphs with Fractional Total Chromatic Number Equal to

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2009.11.039 ◽

2009 ◽

Vol 35 ◽

pp. 235-240 ◽

Cited By ~ 3

Author(s):

Takehiro Ito ◽

W. Sean Kennedy ◽

Bruce A. Reed

Keyword(s):

Chromatic Number ◽

Total Chromatic Number

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ON DYNAMIC COLORING OF WEB GRAPH

Kongunadu Research Journal ◽

10.26524/krj263 ◽

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.

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On r-dynamic coloring of comb graphs

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.2.191-200 ◽

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.

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Total Coloring Conjecture for Certain Classes of Graphs

Algorithms ◽

10.3390/a11100161 ◽

2018 ◽

Vol 11 (10) ◽

pp. 161 ◽

Cited By ~ 1

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.

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