On a problem of J. Zaks concerning 5-valent 3-connected planar graphs (original) (raw)

A sufficient condition for planar graphs to be acyclically 5-choosable

Journal of Graph Theory, 2012

A proper vertex coloring of a graph G = (V , E) is acyclic if G contains no bicolored cycle. Given a list assignment L ={L(v)|v ∈ V } of G, we say G is acyclically L-list colorable if there exists a proper acyclic coloring of G such that (v) ∈ L(v) for all v ∈ V. If G is acyclically L-list colorable for any list assignment with |L(v)|≥ k for all v ∈ V , then G is acyclically k-choosable. In this article we prove that every planar graph without 4-cycles and without intersecting triangles is acyclically 5-choosable. This improves the result in [M. Chen and W. Wang, Discrete Math 308 (2008), 6216-6225], which says that every planar graph without 4-cycles and without two triangles at distance less than 3 is acyclically 5-choosable.

On the Existence of Specific Stars in Planar Graphs

Graphs and Combinatorics, 2007

Given a graph G, a (k; a, b, c)-star in G is a subgraph isomorphic to a star K 1,3 with a central vertex of degree k and three leaves of degrees a, b and c in G. The main result of the paper is:

On 3-simplicial vertices in planar graphs

Discussiones Mathematicae Graph Theory, 2004

A vertex v in a graph G = (V, E) is k-simplicial if the neighborhood N (v) of v can be vertex-covered by k or fewer complete graphs. The main result of the paper states that a planar graph of order at least four has at least four 3simplicial vertices of degree at most five. This result is a strengthening of the classical corollary of Euler's Formula that a planar graph of order at least four contains at least four vertices of degree at most five.

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.

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.

On Almost-Planar Graphs

The Electronic Journal of Combinatorics, 2018

A nonplanar graph GGG is called almost-planar if for every edge eee of GGG, at least one of GbackslasheG\backslash eGbackslashe and G/eG/eG/e is planar. In 1990, Gubser characterized 3-connected almost-planar graphs in his dissertation. However, his proof is so long that only a small portion of it was published. The main purpose of this paper is to provide a short proof of this result. We also discuss the structure of almost-planar graphs that are not 3-connected.

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.

3-Choosability of Triangle-Free Planar Graphs with Constraints on 4-Cycles

SIAM Journal on Discrete Mathematics, 2010

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch [2] asserts that every triangle-free planar graph is 3-colorable. On the other hand Voigt [10] found such a graph which is not 3-choosable. We prove that if a triangle-free planar graph is not 3-choosable, then it contains a 4-cycle that intersects another 4-or 5-cycle in exactly one edge. This strengthens the Thomassen's result [8] that every planar graph of girth at least 5 is 3-choosable. In addition, this implies that every triangle-free planar graph without 6-and 7-cycles is 3-choosable.

Choosability of planar graphs of girth 5

2011

Thomassen proved that any plane graph of girth 5 is list-colorable from any list assignment such that all vertices have lists of size two or three and the vertices with list of size two are all incident with the outer face and form an independent set. We present a strengthening of this result, relaxing the constraint on the vertices with list of size two. This result is used to bound the size of the 3-list-coloring critical plane graphs with one precolored face.

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].