A note on acyclic number of planar graphs (original) (raw)

Large induced forests in planar graphs with girth 4

Discrete Applied Mathematics, 2019

We give here some new lower bounds on the order of a largest induced forest in planar graphs with girth 4 and 5. In particular we prove that a triangle-free planar graph of order n admits an induced forest of order at least 6n+7 11 , improving the lower bound of Salavatipour [M. R. Salavatipour, Large induced forests in trianglefree planar graphs, Graphs and Combinatorics, 22:113-126, 2006]. We also prove that a planar graph of order n and girth at least 5 admits an induced forest of order at least 44n+50 69 .

On the vertex-arboricity of planar graphs

European Journal of Combinatorics, 2008

The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well-known that a(G) ≤ 3 for any planar graph G. In this paper we prove that a(G) ≤ 2 whenever G is planar and either G has no 4-cycles or any two triangles of G are at distance at least 3.

On the vertex-arboricity of planar graphs without 7-cycles

Discrete Mathematics, 2012

The vertex arboricity va(G) of a graph G is the minimum number of colors the vertices can be labeled so that each color class induces a forest. It was well-known that va(G) ≤ 3 for every planar graph G. In this paper, we prove that va(G) ≤ 2 if G is a planar graph without 7-cycles. This extends a result in [A. Raspaud, W. Wang, On the vertex-arboricity of planar graphs, European J. Combin. 29 (2008) 1064-1075] that for each k ∈ {3, 4, 5, 6}, planar graphs G without k-cycles have va(G) ≤ 2.

The Linear 2-Arboricity of Planar Graphs

Graphs and Combinatorics, 2003

Let G be a planar graph with maximum degree D and girth g. The linear 2-arboricity la 2 ðGÞ of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. We prove that (1) la 2 ðGÞ dðD þ 1Þ=2e þ 12; (2) la 2 ðGÞ dðD þ 1Þ=2e þ 6 if g ! 4; (3) la 2 ðGÞ dðD þ 1Þ=2e þ 2 if g ! 5; (4) la 2 ðGÞ dðD þ 1Þ=2e þ 1 if g ! 7.

Planar Graphs Have Exponentially Many 3-Arboricities

SIAM Journal on Discrete Mathematics, 2012

It is well-known that every planar or projective planar graph can be 3-colored so that each color class induces a forest. This bound is sharp. In this paper, we show that there are in fact exponentially many 3-colorings of this kind for any (projective) planar graph. The same result holds in the setting of 3-list-colorings.

Vertex-arboricity of planar graphs without intersecting triangles

European Journal of Combinatorics, 2012

The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which vertex set V (G) can be partitioned so that each subset induces an acyclic graph. In this paper, we prove one of the conjectures proposed by Raspaud and Wang (2008) [15] which says that a(G) = 2 for any planar graph without intersecting triangles.

A note on the acyclic 3-choosability of some planar graphs

Discrete Applied Mathematics, 2010

An acyclic coloring of a graph G is a coloring of its vertices such that : (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v ∈ V (G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an acyclic coloring φ of G such that φ(v) ∈ L(v) for all v ∈ V (G). If G is acyclically L-list colorable for any list assignment L with |L(v)| ≥ k for all v ∈ V (G), then G is acyclically k-choosable. In this paper, we prove that every planar graph with neither cycles of lengths 4 to 7 (resp. to 8, to 9, to 10) nor triangles at distance less 7 (resp. 5, 3, 2) is acyclically 3-choosable.

Structural properties of 1-planar graphs and an application to acyclic edge coloring

2010

A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some new classes of light graphs in 1-planar graphs with the bounded degree are found. Therefore, two open problems presented by Fabrici and Madaras [The structure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] are solved. Furthermore, we prove that each 1-planar graph G with maximum degree ∆(G) is acyclically edge L-choosable where L = max{2∆(G) − 2, ∆(G) + 83}.

An improved lower bound for the planar Tur\'an number of cycles

Cornell University - arXiv, 2022

The planar Turán number of a graph H, denoted by ex P (n, H), is the largest number of edges in a planar graph on n vertices without containing H as a subgraph. In this paper, we continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213-230]. We first obtain an improved lower bound for ex P (n, C k) for all k ≥ 13 and n ≥ 5(k − 6 + ⌊(k − 1)/2⌋)(k − 1)/2; the construction for each k and n provides a simpler counterexample to a conjecture of Ghosh, Győri, Martin, Paulos and Xiao [arXiv:2004.14094v1], which has recently been disproved by Cranston, Lidický, Liu and Shantanam [Electron. J. Combin. 29(3) (2022) #P3.31] for every k ≥ 11 and n sufficiently large (as a function of k). We then prove that ex P (n, H +) = ex P (n, H) for all k ≥ 5 and n ≥ |H| + 1, where H ∈ {C k , 2C k } and H + is obtained from H by adding a pendant edge to a vertex of degree two.