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- W3105697352 abstract "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 $g(x)=lfloorfrac{8}{5}xrfloor$ 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." @default.
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- W3105697352 date "2013-01-01" @default.
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- W3105697352 title "Complements of nearly perfect graphs" @default.
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- W3105697352 doi "https://doi.org/10.4310/joc.2013.v4.n3.a2" @default.
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