Partitioning Sparse Graphs into an Independent Set and a Forest of Bounded Degree

An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum avera...

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Published inThe Electronic journal of combinatorics Vol. 25; no. 1
Main Authors Dross, François, Montassier, Mickael, Pinlou, Alexandre
Format Journal Article
LanguageEnglish
Published Open Journal Systems 02.03.2018
Subjects
Combinatorics
Computer Science
Discrete Mathematics
Mathematics
Vertex decompositions
Forests
Graph theory
Sparse graphs
Planar graphs
Independent sets
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Summary:An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum average degree less than $M$, then it has an $({\cal I},{\cal F}_d)$-partition. Additionally, we prove that for all $\frac{8}{3} \le M < 3$ and $d \ge \frac{1}{3-M}$, if a graph has maximum average degree less than $M$ then it has an $({\cal I},{\cal F}_d)$-partition. It follows that planar graphs with girth at least $7$ (resp. $8$, $10$) admit an $({\cal I},{\cal F}_5)$-partition (resp. $({\cal I},{\cal F}_3)$-partition, $({\cal I},{\cal F}_2)$-partition).
ISSN:1077-8926
1077-8926
DOI:10.37236/6815