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Most direct product of graphs are Type 1 [article]

Diane Castonguay and Celina M. H. de Figueiredo and Luis Antonio Kowada and Caroline Reis Patrão and Diana Sasaki
2021 arXiv   pre-print
Most classified direct product of graphs are Type 1.  ...  A k-total coloring of a graph G is an assignment of k colors to its elements (vertices and edges) so that adjacent or incident elements have different colors.  ...  Acknowledgements This work is partially supported by the Brazilian agencies CNPq (Grant numbers: 302823/2016-6, 407635/2018-1 and 313797/2020-0) and FAPERJ (Grant numbers: CNE E-26/202.793/2017 and ARC  ... 
arXiv:2110.14453v1 fatcat:othwnab6s5gbrm2bht34rdexti

Page 8803 of Mathematical Reviews Vol. , Issue 2002M [page]

2002 Mathematical Reviews  
This paper discusses the fractional chromatic number of the direct product (also known as Kronecker product, tensor product, categorical product and cardinal product) of graphs.  ...  A proper total coloring of a multigraph that is simultaneously a p-coloring of the incidentors of G is called a total p-coloring.  ... 

Page 6041 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews  
products of graphs.  ...  The exact total chromatic number of any balanced complete r-partite graph has been determined.  ... 

Page 600 of Mathematical Reviews Vol. 43, Issue 3 [page]

1972 Mathematical Reviews  
Compute the product F,F,---Fy_, as a sum of products of edges and in this computation delete (i.e., replace by zero) any product containing a set of edges that forms a directed circuit.  ...  The author studies the minimum number of intersections of edges when realizing a complete k-partite graph K,,,,.  ... 

Some problems on idomatic partitions and b-colorings of direct products of complete graphs

Mario Valencia-Pabon
2010 Matemática contemporânea  
We recall some known results on idomatic partitions of direct products of complete graphs and we present new results concerning the b-colorings of the direct product of two complete graphs.  ...  In this note, we deal with the characterization of the idomatic partitions and b-colorings of direct products of complete graphs.  ...  Recently, in [9] it is given a full characterization of the idomatic partitions of the direct product of three complete graphs.  ... 
doi:10.21711/231766362010/rmc3911 fatcat:a6bs7vazwbebdcaquzghgig7re

Total Colourings - A survey [article]

Geetha Jayabalan and Narayanan N and K Somasundaram
2018 arXiv   pre-print
In this paper we give a survey on total coloring of graphs.  ...  The smallest integer k needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph.  ...  Geetha and Somasundaram [39] proved that direct product of two even complete graphs are type-I and the direct product of two cycles C m and C n are type-I for certain values of m and n.  ... 
arXiv:1812.05833v1 fatcat:2igatxvf55bcncqp2q2riwpp2y

Parameterizations of hitting set of bundles and inverse scope

Peter Damaschke
2013 Journal of combinatorial optimization  
The problem has a graph-theoretic formulation as a reachability problem in directed graphs.  ...  On the positive side, we give an FPT algorithm where the parameter is the total number of compounds involved in the reactions.  ...  The author would like to thank the referees for valuable comments that made the presentation clearer, and for suggesting the Turing way to show W [1] membership.  ... 
doi:10.1007/s10878-013-9629-5 fatcat:6a7icbjphfc5blkr2yy67rocia

Page 1537 of Mathematical Reviews Vol. 46, Issue 6 [page]

1973 Mathematical Reviews  
Tomova, Eliska 8894 On the decomposition of the complete directed graph into factors with given diameters. Mat. Casopis Sloven. Akad. Vied 20 (1970), 257-261.  ...  The lexicographic product D o E of directed graphs D and Fis an associative noncommutative operation, as it is for graphs.  ... 

Page 2783 of Mathematical Reviews Vol. , Issue 85g [page]

1985 Mathematical Reviews  
In particular, we consider the induced subgraphs of an infinite graph, relabelling of its vertices, the complementary graph, the union, Cartesian product, complete product and direct sum of two infinite  ...  For some of these operations we find that the spectrum of the graph so obtained is always infinite (direct sum, line and total graph).  ... 

A new graph over semi-direct products of groups

Sercan Topkaya, Sinan Cevik
2016 Filomat  
In this paper, by establishing a new graph Γ(G) over the semi-direct product of groups, we will first state and prove some graph-theoretical properties, namely, diameter, maximum and minimum degrees, girth  ...  In the final section we will show that Γ(G) is actually a perfect graph.  ...  Acknowledgement The authors are grateful to the referees for comments and corrections that have helped to improve the text.  ... 
doi:10.2298/fil1603611t fatcat:2r6q2slurbb5jgxstcdmupdlnq

Page 42 of Mathematical Reviews Vol. , Issue 94a [page]

1994 Mathematical Reviews  
{For the entire collection see MR 93m:00040.} 94a:05079 05C15 Zhang, Zhong Fu (PRC-LRI; Lanzhou); Wang, Jian Fang (PRC-ASBJ-AM; Beijing) A summary of the progress on total colorings of graphs.  ...  94a:05078 every k-chromatic graph can be contracted to a complete graph on k vertices. In order to prove it, G. A.  ... 

The b-Chromatic Number of Cubic Graphs

Marko Jakovac, Sandi Klavžar
2010 Graphs and Combinatorics  
We prove that a graph with every edge belonging to at most c cycles has game coloring number of at most c + 4.  ...  We propose a generalization of the Fibonacci numbers, the Lucas numbers, the Pell numbers, the Pell-Lucas numbers, the Tribonacci numbers and next we give their graph representations with respect to the  ...  INTERVAL EDGE COLORINGS OF SOME PRODUCTS STAR COLORING OF GRAPHS Andre Raspaud Universite Bordeaux 1, Talence, France A proper coloring of the vertices of a graph is called a star coloring if the union  ... 
doi:10.1007/s00373-010-0898-9 fatcat:5j4t3ji33bbopoker7veh5aybi

Snarks with total chromatic number 5

Gunnar Brinkmann, Myriam Preissmann, Diana Sasaki
2015 Discrete Mathematics & Theoretical Computer Science  
The total chromatic number of G, denoted by χT(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5.  ...  Graph Theory International audience A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors.  ...  Acknowledgments This research was conducted using the resources of the Stevin supercomputer Infrastructure at Ghent University. Without these resources some results could not have been obtained.  ... 
doi:10.46298/dmtcs.2111 fatcat:ghmwdb4ykfbsddtzxayit6qswq

Total Coloring Conjecture for Certain Classes of Graphs

R. Vignesh, J. Geetha, K. Somasundaram
2018 Algorithms  
In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.  ...  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.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/a11100161 fatcat:xyb5t4gmkjhm3m2jndd5rhqs2u

Page 7695 of Mathematical Reviews Vol. , Issue 2001K [page]

2001 Mathematical Reviews  
The minimum number of colours necessary for total p-colouring of a directed multigraph G is called the total p-step chromatic number of G and it is denoted by 7‘(p,G).  ...  Summary: “For a graph G and an order ao on V(G), we define a greedy defining set as a subset S of V(G) with an assignment of colors to vertices in S such that the pre-coloring can be extended to a x(G)  ... 
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