scholarly journals Perfect graphs of arbitrarily large clique-chromatic number

2016 ◽  
Vol 116 ◽  
pp. 456-464 ◽  
Author(s):  
Pierre Charbit ◽  
Irena Penev ◽  
Stéphan Thomassé ◽  
Nicolas Trotignon
Keyword(s):  
Chromatic Number ◽  
Perfect Graphs ◽  
Large Clique

scholarly journals On the Unitary Cayley Graph of a Finite Ring

10.37236/206 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Reza Akhtar ◽  
Megan Boggess ◽  
Tiffany Jackson-Henderson ◽  
Isidora Jiménez ◽  
Rachel Karpman ◽  
...  
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Finite Ring ◽  
Perfect Graphs ◽  
Edge Connectivity ◽  
Unitary Cayley Graph

We study the unitary Cayley graph associated to an arbitrary finite ring, determining precisely its diameter, girth, eigenvalues, vertex and edge connectivity, and vertex and edge chromatic number. We also compute its automorphism group, settling a question of Klotz and Sander. In addition, we classify all planar graphs and perfect graphs within this class.

Computing clique and chromatic number of circular-perfect graphs in polynomial time

2012 ◽  
Vol 141 (1-2) ◽  
pp. 121-133 ◽  
Author(s):  
Arnaud Pêcher ◽  
Annegret K. Wagler
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Perfect Graphs

Graphs with large clique-chromatic numbers

2015 ◽  
Vol 07 (04) ◽  
pp. 1550055
Author(s):  
Tanawat Wichianpaisarn ◽  
Chariya Uiyyasathian
Keyword(s):  
Chromatic Number ◽  
Maximal Clique ◽  
Line Graphs ◽  
Chromatic Numbers ◽  
Bounded Set ◽  
Initial Graph ◽  
The Family ◽  
Large Clique

The clique-chromatic number of a graph [Formula: see text], [Formula: see text], is the least number of colors on [Formula: see text] without a monocolored maximal clique of size at least two. If [Formula: see text] is triangle-free, [Formula: see text]; we then consider only graphs with a triangle. Unlike the chromatic number, the clique-chromatic number of a graph is not necessary to be at least those of its subgraphs. Thus, for any family of graphs [Formula: see text], the boundedness of [Formula: see text][Formula: see text] has been investigated. Many families of graphs are proved to have a bounded set of clique-chromatic numbers. In literature, only few families of graphs are shown to have an unbounded set of clique-chromatic numbers, for instance, the family of line graphs. This paper gives another family of graphs with such an unbounded set. These graphs are obtained by the well-known Mycielski’s construction with a certain property of the initial graph.

scholarly journals The Guessing Number of Undirected Graphs

10.37236/679 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Demetres Christofides ◽  
Klas Markström
Keyword(s):  
Network Coding ◽  
Chromatic Number ◽  
Lower Bounds ◽  
Perfect Graphs ◽  
Upper And Lower Bounds ◽  
Undirected Graphs ◽  
Fractional Chromatic Number ◽  
Winning Probability ◽  
Optimal Bounds

Riis [Electron. J. Combin., 14(1):R44, 2007] introduced a guessing game for graphs which is equivalent to finding protocols for network coding. In this paper we prove upper and lower bounds for the winning probability of the guessing game on undirected graphs. We find optimal bounds for perfect graphs and minimally imperfect graphs, and present a conjecture relating the exact value for all graphs to the fractional chromatic number.

Clique and chromatic number of circular-perfect graphs

2010 ◽  
Vol 36 ◽  
pp. 199-206 ◽  
Author(s):  
Arnaud Pêcher ◽  
Annegret K. Wagler
Keyword(s):  
Chromatic Number ◽  
Perfect Graphs

scholarly journals WEAKLY PERFECT GRAPHS ARISING FROM RINGS

2010 ◽  
Vol 52 (3) ◽  
pp. 417-425 ◽  
Author(s):  
H. R. MAIMANI ◽  
M. R. POURNAKI ◽  
S. YASSEMI
Keyword(s):  
Explicit Formula ◽  
Chromatic Number ◽  
Clique Number ◽  
Perfect Graphs ◽  
New Class

AbstractA graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given.

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