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Article Dans Une Revue The Electronic Journal of Combinatorics Année : 2013

Complements of nearly perfect graphs

Raphael Machado
  • Fonction : Auteur
András Sebő
  • Fonction : Auteur
  • PersonId : 739877
  • IdHAL : sebo
Stéphan Thomassé
Nicolas Trotignon
OC

Résumé

A class of graphs closed under taking induced subgraphs is $\chi$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $\chi(G) \leq f(\omega(G))$. We consider the following question initially studied in [A. Gy{\'a}rf{\'a}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $\chi$-bounded class $\cal C$, is the class $\bar{C}$ $\chi$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $\chi$-bounded by the constant function $f(x)=3$, then $\bar{\cal C}$ is $\chi$-bounded by $g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for every constant $c>0$, if $\cal C$ is $\chi$-bounded by a function $f$ such that $f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $\chi$-bounded. For every $j$, we construct a class of graphs $\chi$-bounded by $f(x)=x+x/\log^j(x)$ whose complement is not $\chi$-bounded.

Dates et versions

hal-01592744 , version 1 (25-09-2017)

Identifiants

Citer

Andras Gyarfas, Zhentao Li, Raphael Machado, András Sebő, Stéphan Thomassé, et al.. Complements of nearly perfect graphs. The Electronic Journal of Combinatorics, 2013, 4 (3), pp.299 - 310. ⟨10.4310/JOC.2013.v4.n3.a2⟩. ⟨hal-01592744⟩
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