The chromatic class of a multigraph

Cybernetics ◽  
1965 ◽  
Vol 1 (3) ◽  
pp. 32-41 ◽  
Author(s):  
V. G. Vizing
Keyword(s):  
Chromatic Class

Graphs which are vertex‐critical with respect to the edge‐chromatic class

Mathematika ◽  
1989 ◽  
Vol 36 (2) ◽  
pp. 241-252 ◽  
Author(s):  
A. J. W. Hilton ◽  
P. D. Johnson
Keyword(s):  
Chromatic Class

scholarly journals Rainbow-free $3$-colorings of Abelian Groups

10.37236/2054 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Amanda Montejano ◽  
Oriol Serra
Keyword(s):  
Abelian Group ◽  
Arithmetic Progression ◽  
Structural Characterization ◽  
Abelian Groups ◽  
Cyclic Groups ◽  
Chromatic Class

A $3$-coloring of the elements of an abelian group is said to be rainbow-free if there is no $3$-term arithmetic progression with its members having pairwise distinct colors. We give a structural characterization of rainbow-free colorings of abelian groups. This characterization proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow-free $3$-coloring of cyclic groups.

scholarly journals The Diachromatic Number of Digraphs

10.37236/7807 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Gabriela Araujo-Pardo ◽  
Juan José Montellano-Ballesteros ◽  
Mika Olsen ◽  
Christian Rubio-Montiel
Keyword(s):  
Chromatic Number ◽  
Directed Graphs ◽  
Interpolation Property ◽  
Acyclic Coloring ◽  
Chromatic Class

We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromaticnumber is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichromatic and diachromatic numbers are, respectively, the smallest and the largest number of colors in a complete acyclic coloring. We give some general results for the diachromatic number and study it for tournaments. We also show that the interpolation property for complete acyclic colorings does hold and establish Nordhaus-Gaddum relations.

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