Related papers: Induced Forests in Bipartite Planar Graphs
Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related…
We initiate a systematic study of the fractional vertex-arboricity of planar graphs and demonstrate connections to open problems concerning both fractional coloring and the size of the largest induced forest in planar graphs. In particular,…
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently,…
The paper is devoted to sufficient conditions for the existence of vertex cuts in simple graphs, where the induced subgraph on the cut vertices belongs to a specified graph class. In particular, we show that any connected graph with $n$…
We propose the conjecture that every graph $G$ of order $n$ with less than $3n-6$ edges has a vertex cut that induces a forest. Maximal planar graphs do not have such vertex cuts and show that the density condition would be best possible.…
We show that every graph has an induced pseudoforest of at least $n-m/4.5$ vertices, an induced partial 2-tree of at least $n-m/5$ vertices, and an induced planar subgraph of at least $n-m/5.2174$ vertices. These results are constructive,…
We prove that every triangle-free planar graph of order $n$ and size $m$ has an induced linear forest with at least $\frac{9n - 2m}{11}$ vertices, and thus at least $\frac{5n + 8}{11}$ vertices. Furthermore, we show that there are…
An equitable partition of a graph $G$ is a partition of the vertex-set of $G$ such that the sizes of any two parts differ by at most one. We show that every graph with an acyclic coloring with at most $k$ colors can be equitably partitioned…
In this paper, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2-connected subcubic graph on $n$ vertices and $m$ edges with…
A long-standing conjecture by Albertson and Berman states that every planar graph of order $n$ has an induced forest with at least $\lceil \frac{n}{2} \rceil$ vertices. As a variant of this conjecture, Chappell conjectured that every planar…
A graph is near-bipartite if its vertex set can be partitioned into an independent set and a set which induces a forest. In this paper, planar graphs without cycles of length from 4 to 7 are shown to be near-bipartite.
It is proved that the vertex set of any simple graph $G$ can be equitably partitioned into $k$ subsets for any integer $k\geq\max\{\big\lceil\frac{\Delta(G)+1}{2}\big\rceil,\big\lceil\frac{|G|}{4}\big\rceil\}$ so that each of them induces a…
Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph $G$ can be equitably partitioned into $\lceil(\Delta(G)+1)/2\rceil$ subsets so that each of them induces a forest of $G$. In this note, we prove this conjecture…
Graph isomorphism, subgraph isomorphism, and maximum common subgraphs are classical well-investigated objects. Their (parameterized) complexity and efficiently tractable cases have been studied. In the present paper, for a given set of…
Consider the following process on a simple graph without isolated vertices: Order the edges randomly and keep an edge if and only if it contains a vertex which is not contained in some preceding edge. The resulting set of edges forms a…
Chernyshev, Rauch, and Rautenbach proved that every connected graph on $n$ vertices with less than $\frac{11}{5}n-\frac{18}{5}$ edges has a vertex cut that induces a forest, and conjectured that the same remains true if the graph has less…
An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs…
In this paper we investigate the bipartite analogue of the strong Erdos-Hajnal property. We prove that for every forest $H$ and every $\tau>0$ there exists $\epsilon>0$, such that if $G$ has a bipartition $(A,B)$ and does not contain $H$ as…
We strengthen a result of Dross, Montassier and Pinlou (2017) that the vertex set of every triangle-free planar graph can be decomposed into a set that induces a forest and a set that induces a forest with maximum degree at most $5$,…
This exposition contains a short and streamlined proof of the recent result of Kwan, Letzter, Sudakov and Tran that every triangle-free graph with minimum degree $d$ contains an induced bipartite subgraph with average degree $\Omega(\ln…