scholarly journals A proof of Mader's conjecture on large clique subdivisions in C4-free graphs

2017 ◽  
Vol 95 (1) ◽  
pp. 203-222 ◽  
Author(s):  
Hong Liu ◽  
Richard Montgomery
Keyword(s):  
Large Clique

scholarly journals Subdivisions of a large clique in C6-free graphs

2015 ◽  
Vol 112 ◽  
pp. 18-35 ◽  
Author(s):  
József Balogh ◽  
Hong Liu ◽  
Maryam Sharifzadeh
Keyword(s):  
Large Clique

scholarly journals Finding Large Clique Minors is Hard

2009 ◽  
Vol 13 (2) ◽  
pp. 197-204 ◽  
Author(s):  
David Eppstein
Keyword(s):  
Large Clique

scholarly journals Large Cliques in a Power-Law Random Graph

2010 ◽  
Vol 47 (4) ◽  
pp. 1124-1135 ◽  
Author(s):  
Svante Janson ◽  
Tomasz Łuczak ◽  
Ilkka Norros
Keyword(s):  
Random Graph ◽  
Power Law ◽  
High Probability ◽  
Simple Algorithm ◽  
Heavy Tail ◽  
Degree Sequences ◽  
Constant Size ◽  
Tail Distribution ◽  
Law Degree ◽  
Large Clique

In this paper we study the size of the largest clique ω(G(n, α)) in a random graph G(n, α) on n vertices which has power-law degree distribution with exponent α. We show that, for ‘flat’ degree sequences with α > 2, with high probability, the largest clique in G(n, α) is of a constant size, while, for the heavy tail distribution, when 0 < α < 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm with high probability finds in G(n, α) a large clique of size (1 − o(1))ω(G(n, α)) in polynomial time.

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.

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