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A Bound on the Total Chromatic Number

1998
*
Combinatorica
*

In particular, we show that for su ciently large,

doi:10.1007/pl00009820
fatcat:q2yt4mkiuzbhxj3a3zjkgki6rm
*the**total**chromatic**number*of such*a*graph is at most + 10 26 .*The*proof is probabilistic. ... We prove that*the**total**chromatic**number*of any graph with maximum degree is at most plus an absolute constant. ...*The**total**chromatic**number*, 00 (G) is*the*least*number*of colours required for*a**total*colouring of G. ...##
###
A new upper bound for total colourings of graphs

1995
*
Discrete Mathematics
*

We give

doi:10.1016/0012-365x(94)00219-9
fatcat:4hruj5izt5h7tlcfinnred3rby
*a*new upper*bound**on**the**total**chromatic**number*of*a*graph. This*bound*improves*the*results known for some classes of graphs. ...*The**bound*is stated as follows: ZT ~< Z~ + L l3 ~ J + 2, where Z is*the**chromatic**number*, Z~ is*the*edge*chromatic**number*(*chromatic*index) and ZT is*the**total**chromatic**number*. ...*The**chromatic**number*•(G), edge*chromatic**number*ze(G),*total**chromatic**number*~T(G) is*the*least*number*of colours in*a*vertex, edge,*total*colouring of G, respectively. ...##
###
Local super antimagic total face coloring of planar graphs

2019
*
IOP Conference Series: Earth and Environment
*

Acknowledgement We are grateful for

doi:10.1088/1755-1315/243/1/012117
fatcat:uiw7yu5qwfcm3g5quohtttgl5a
*a*wonderful support from Universitas Islam Negeri Syarif Hidayatullah Jakarta and CGANT -University of Jember in 2018. ... We combine an upper*bound*value with*a*lower*bound*value. We get γ latf (L n ) = 2. 2.4.*The*local super antimagic*total*face*chromatic**number*of circular ladder graph Let n ≥ 3 be*a*natural*number*. ...*The*local super antimagic*total*face*chromatic**number*of ladder graph Let n ≥ 2 be*a*natural*number*. ...##
###
Page 53 of Mathematical Reviews Vol. , Issue 2002A
[page]

2002
*
Mathematical Reviews
*

*On*

*the*

*total*

*chromatic*

*number*of Steiner systems. ... Summary: “In this work we give

*a*new lower

*bound*

*on*

*the*

*chromatic*

*number*of

*a*Mycielski graph M;. ...

##
###
Page 48 of Mathematical Reviews Vol. , Issue 2000a
[page]

2000
*
Mathematical Reviews
*

We also show that

*the**total*choice*number*of*a*graph of maximum degree 2 is equal to its*total**chromatic**number*.” 2000a:05082 05C15 Kayll, P. ... Enomoto, Hornak, and Jendrol’ [“Cyclic*chromatic**number*of 3-connected plane graphs”, submitted] obtained*the*above sharp*bound*under weaker restrictions*on*k. ...##
###
Tight bounds on the chromatic sum of a connected graph

1989
*
Journal of Graph Theory
*

In this article we determine tight

doi:10.1002/jgt.3190130310
fatcat:rrkc277lt5bm3hte2wld426ita
*bounds**on**the**chromatic*sum of*a*connected graph with e edges, ...*The**chromatic*sum of*a*graph is introduced in*the*dissertation of Ewa Kubicka. It is*the*smallest possible*total*among all proper colorings of G using natural*numbers*. ... ACKNOWLEDGMENT*The*research of P. J. M. and*A*. J. S. was supported in part by*the*Office of Naval Research Contract N00014-86-K-0566. ...##
###
Two-floor buildings need eight colors

2015
*
Journal of Graph Algorithms and Applications
*

We also discuss

doi:10.7155/jgaa.00344
fatcat:oulq75r4zjeorjonqj7sqwr3wi
*bounds**on**the**chromatic**number*of*the*adjacency graph of general arrangements of 3-dimensional parallelepipeds according to geometrical measures of*the*parallelepipeds (side length,*total*... In*the*case each parallelepiped is within*one*floor,*a*direct application of*the*Four-Colour Theorem yields that*the*adjacency graph has*chromatic**number*at most 8. ...*The*third author thanks Louigi Addario-Berry, Frédéric Havet, Ross Kang, Colin McDiarmid and Tobias Müller for stimulating discussions*on**the*topic of this paper during*a*stay in Oxford, in 2005. ...##
###
Total Dominator Colorings In Paths
[article]

2012
*
Zenodo
*

Determining

doi:10.5281/zenodo.9127
fatcat:3jgga2tdbnhzbbufuyjcr2io3e
*the**total*dominator*chromatic**number*in paths. ... Next,we obtain*a**bound*for td-*chromatic**number*through*the**total*domination*number*and*chromatic**number*. Theorem 2.7.Let G be any graph with (G) 1. ...*The**total*dominator*chromatic**number*(td-*chromatic**number*) of G is defined as minimum*number*of colors needed in*a**total*dominator coloring of G and is denoted by td (G). ...##
###
Page 4149 of Mathematical Reviews Vol. , Issue 93h
[page]

1993
*
Mathematical Reviews
*

For k = 3,

*the*k-cyclic*chromatic**number*coincides with*the**chromatic**number*of*the*graph. For k = 4, it is at most 6 and this upper*bound*is tight. ... It is proved that*a*graph G with 2n vertices and maximum degree 2n—2 has*total**chromatic**number*27 — 1 unless*the*complement of G is K2U Kj 2n-3, in which case*the**total**chromatic**number*is 27. ...##
###
Page 6090 of Mathematical Reviews Vol. , Issue 2001I
[page]

2001
*
Mathematical Reviews
*

*The*lower

*bound*for

*the*

*chromatic*

*number*of R” has been (1.207+ 0(1))".

*The*paper improves

*the*lower

*bound*to (1.239 + o(1))”. ...

*On*

*the*other hand, for every

*number*f; > fo,

*the*interval from fp to ¢; contains

*a*

*chromatic*root of

*a*graph with

*a*Hamiltonian path. ...

##
###
Two floor building needing eight colors
[article]

2014
*
arXiv
*
pre-print

We also discuss

arXiv:1405.6620v1
fatcat:iejlsbhfibcybalo4up3u35zai
*bounds**on**the**chromatic**number*of*the*adjacency graph of general arrangements of 3-dimensional parallelepipeds according to geometrical measures of*the*parallelepipeds (side length,*total*... In*the*case each parallelepiped is within*one*floor,*a*direct application of*the*Four-Colour Theorem yields that*the*adjacency graph has*chromatic**number*at most 8. ...*The*third author thanks Louigi Addario-Berry, Frédéric Havet, Ross Kang, Colin McDiarmid and Tobias Müller for stimulating discussions*on**the*topic of this paper during*a*stay in Oxford, in 2005. ...##
###
Clique Number vs. Chromatic Number in Wireless Interference Graphs: Simulation Results

2007
*
IEEE Communications Letters
*

In our extensive simulations with UDGs of random networks, we observed that

doi:10.1109/lcomm.2007.070257
fatcat:vskujgvgnbaeni5kudefbzqhsq
*the*clique*number*and*the**chromatic**number*values were typically very close to*one*another and*the*maximum deviation was much ... less than*the*theoretical*bounds*. ... While*the**chromatic**number*problem*on**a*UDG is still NP-complete,*the*clique*number*problem can be solved in polynomial time in UDGs [3]. ...##
###
Page 5292 of Mathematical Reviews Vol. , Issue 93j
[page]

1993
*
Mathematical Reviews
*

*The*

*bounded*

*chromatic*

*number*7;(G) of G is

*the*smallest

*number*of colors such that G admits

*a*

*bounded*coloring. ... >n, then

*the*

*total*

*chromatic*

*number*y” < y'+k+1, where x’ is

*the*edge-

*chromatic*

*number*of G. ...

##
###
Page 6613 of Mathematical Reviews Vol. , Issue 99j
[page]

1999
*
Mathematical Reviews
*

(F-PARIS6-CM; Paris)

*A**bound**on**the**total**chromatic**number*. (English summary) Combinatorica 18 (1998), no. 2, 241-280. ...*The**total**chromatic**number*of G is*the*least*number*of colours required for*a**total*colouring of G. ...##
###
Chromatic Number and the 2-Rank of a Graph

2001
*
Journal of combinatorial theory. Series B (Print)
*

We show that if

doi:10.1006/jctb.2000.2003
fatcat:w2buzib4nbdltl5y2wwgm5bx7q
*the*adjacency matrix of*a*graph X has 2-rank 2r, then*the**chromatic**number*of X is at most 2 r +1, and that this*bound*is tight. ... Kotlov and Lovasz [6]*bound**the**number*of vertices of*a*reduced graph in terms of*the*rank, and Kotlov [7] uses this*bound*to get an improved*bound**on**the**chromatic**number*in terms of*the*rank. ...*CHROMATIC**NUMBER*To complete our main result, we merely need to establish*the**chromatic**number*of Y(2r). We start by finding*a**bound**on**the*size of an independent set in Y(2r). ...
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