On Light Graphs in 3-Connected Plane Graphs Without Triangular or Quadrangular Faces (original) (raw)

On 3-Connected Plane Graphs without Triangular Faces

Journal of Combinatorial Theory, Series B, 1999

We prove that each polyhedral triangular face free map G on a compact 2dimensional manifold M with Euler characteristic (M) contains a k-path, i.e. a path on k vertices, such that each vertex of this path has, in G, degree at most 5 2 k if M is a sphere S 0 and at most k 2 5+ p 49?24 (M 2 if M 6 = S 0 or does not contain any k-path. We show that for even k this bound is best possible. Moreover, we verify that for any graph other than a path no similar estimation exists.

Light subgraphs of graphs embedded in the plane and in the projective plane – a survey –

2011

It is well known that every planar graph contains a vertex of degree at most 5. A theorem of Kotzig states that every 3-connected planar graph contains an edge whose endvertices have degree-sum at most 13. Recently, Fabrici and Jendrol’ proved that every 3-connected planar graph G that contains a k-vertex path, a path on k vertices, contains also a k-vertex path P such that every vertex of P has degree at most 5k. A result by Enomoto and Ota says that every 3-connected planar graph G of order at least k contains a connected subgraph H of order k such that the degree sum of vertices of H in G is at most 8k− 1. Motivated by these results, a concept of light graphs has been introduced. A graph H is said to be light in a family G of graphs if at least one member of G contains a copy of H and there is an integer w(H,G) such that each member G of G with a copy of H also has a copy K of H with degree sum ∑ v∈V (K) degG(v) ≤ w(H,G). In this paper we present a survey of results on light grap...

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 vertex types and cyclic colourings of 3-connected plane graphs

Discrete Mathematics, 2000

We prove a structural theorem of Lebesgue's type concerning the existence of certain types of vertices in 3-connected plane graphs. This theorem is then applied to the proof such that the cyclic chromatic number c(G)6k + 2 for k = max{40; * (G)}, where * (G) denotes the size of the largest face of G. This conÿrms a conjecture by Plummer and Toft for * (G)¿40.

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 ?

On doubly light vertices in plane graphs

Discussiones Mathematicae Graph Theory, 2011

A vertex is said to be doubly light in a family of plane graphs if its degree and sizes of neighbouring faces are bounded above by a finite constant. We provide several results on the existence of doubly light vertices in various families of plane graph.

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.

Planar Graphs Without Cycles of Specific Lengths

European Journal of Combinatorics, 2002

It is easy to see that planar graphs without 3-cycles are 3-degenerate. Recently, it was proved that planar graphs without 5-cycles are also 3-degenerate. In this paper it is shown, more surprisingly, that the same holds for planar graphs without 6-cycles.