On 3-simplicial vertices in planar graphs (original) (raw)

Note on 3-choosability of planar graphs with maximum degree 4

Discrete Mathematics

Deciding whether a planar graph (even of maximum degree 4) is 3-colorable is NP-complete. Determining subclasses of planar graphs being 3-colorable has a long history, but since Grötzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prove that every planar graph obtained as a subgraph of the medial graph of any bipartite plane graph is 3-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence representing an entirely different class than the graphs inferred by the above mentioned conjectures.

Subgraphs with Restricted Degrees of Their Vertices in Planar 3-Connected Graphs

Graphs and Combinatorics, 1997

We have proved that every 3-connected planar graph G either contains a path on k vertices each of which has degree at most 5k or does not contain any path on k vertices; the bound 5k is the best possible. Moreover, for every connected planar graph H other than a path and for every integer m~3 there is a 3-connected planar graph G such that each copy of H in G contains a vertex of degree at least m.

On a problem of J. Zaks concerning 5-valent 3-connected planar graphs

Discrete Mathematics, 1984

Recently J. Zaks formulated the following Eberhard-type problem: Let (Ps, P6 .... ) be a finite sequence of nonnegative integers; does there exist a 5-valent 3-connected planar graph G such that it has exactly Pk k-gons for all k ~> 5, m i of its vertices meet exactly i triangles, 4 ~< i <~ 5, and m4+2ms=24+3 ~ (k-4)pk ?

Some structural properties of planar graphs and their applications to 3-choosability

Discrete Mathematics, 2012

In this article, we consider planar graphs in which each vertex is not incident to some cycles of given lengths, but all vertices can have different restrictions. This generalizes the approach based on forbidden cycles which corresponds to the case where all vertices have the same restrictions on the incident cycles. We prove that a planar graph G is 3-choosable if it is satisfied one of the following conditions: (1) each vertex x is neither incident to cycles of lengths 4, 9, ix with ix ∈ {5, 7, 8}, nor incident to 6-cycles adjacent to a 3-cycle. (2) each vertex x is not incident to cycles of lengths 4, 7, 9, ix with ix ∈ {5, 6, 8}. This work implies five results already published [13, 3, 7, 12, 4].

Simple k-planar graphs are simple (k + 1)-quasiplanar

Journal of Combinatorial Theory, Series B, 2019

A simple topological graph is k-quasiplanar (k ≥ 2) if it contains no k pairwise crossing edges, and k-planar if no edge is crossed more than k times. In this paper, we explore the relationship between k-planarity and k-quasiplanarity to show that, for k ≥ 2, every k-planar simple topological graph can be transformed into a (k + 1)-quasiplanar simple topological graph.

On the simplicial 3-polytopes with only two types of edges

Discrete Mathematics, 1984

For some families of graphs of simplicial 3-polytopes with two types of edges structural properties are described, for other ones their cardinality is determined. Griinbaum and Motzkin [3], Griinbaum and Zaks [4], and Malkevitch [6] investigated the structural properties of trivalent planar graphs with at most two types of faces. It seems that the knowledge of the structure of such graphs is useful for other reasons as well (cf., e.g. Griinbaum [1, 2], Jucovi~ [5], Owens [7], Zaks [8]). The dual problem may be formulated as follows: Characterize simplicial planar graphs with at most two types of vertices. (A planar graph is simplicial if all its faces are triangles.)

A note on acyclic number of planar graphs

Ars Mathematica Contemporanea

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) ≥ 1 − 3 2g 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) ≥ 1 − 3 g n holds. In addition, we give a construction showing that the constant 3 2 from the conjecture cannot be decreased.

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.

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.