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46. A note on acyclic number of planar graphsMirko Petruševski, Riste Škrekovski, 2017, original scientific article Abstract: The acyclic number ▫$a(G)$▫ of a graph ▫$G$▫ is the maximum order of an induced forest in ▫$G$▫. The purpose of this short paper is to propose a conjecture that ▫$a(G)\geq \left( 1-\frac{3}{2g}\right)n$▫ holds for every planar graph ▫$G$▫ of girth ▫$g$▫ and order ▫$n$▫, which captures three known conjectures on the topic. In support of this conjecture, we prove a weaker result that ▫$a(G)\geq \left( 1-\frac{3}{g} \right)n$▫ holds. In addition, we give a construction showing that the constant ▫$\frac{3}{2}$▫ from the conjecture cannot be decreased.
Keywords: induced forest, acyclic number, planar graph, girth
Published in RUP: 03.01.2022; Views: 993; Downloads: 16
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