Displaying similar documents to “4-chromatic Koester graphs”

Light Graphs In Planar Graphs Of Large Girth

Peter Hudák, Mária Maceková, Tomáš Madaras, Pavol Široczki (2016)

Discussiones Mathematicae Graph Theory

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A graph H is defined to be light in a graph family 𝒢 if there exist finite numbers φ(H, 𝒢) and w(H, 𝒢) such that each G ∈ 𝒢 which contains H as a subgraph, also contains its isomorphic copy K with ΔG(K) ≤ φ(H, 𝒢) and ∑x∈V(K) degG(x) ≤ w(H, 𝒢). In this paper, we investigate light graphs in families of plane graphs of minimum degree 2 with prescribed girth and no adjacent 2-vertices, specifying several necessary conditions for their lightness and providing sharp bounds on φ and w...

Characterizations of planar plick graphs

V.R. Kulli, B. Basavanagoud (2004)

Discussiones Mathematicae Graph Theory

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In this paper we present characterizations of graphs whose plick graphs are planar, outerplanar and minimally nonouterplanar.

On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs

Július Czap, Jakub Przybyło, Erika Škrabuľáková (2016)

Discussiones Mathematicae Graph Theory

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A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same...

2-distance 4-colorability of planar subcubic graphs with girth at least 22

Oleg V. Borodin, Anna O. Ivanova (2012)

Discussiones Mathematicae Graph Theory

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The trivial lower bound for the 2-distance chromatic number χ₂(G) of any graph G with maximum degree Δ is Δ+1. It is known that χ₂ = Δ+1 if the girth g of G is at least 7 and Δ is large enough. There are graphs with arbitrarily large Δ and g ≤ 6 having χ₂(G) ≥ Δ+2. We prove the 2-distance 4-colorability of planar subcubic graphs with g ≥ 22.

1-factors and characterization of reducible faces of plane elementary bipartite graphs

Andrej Taranenko, Aleksander Vesel (2012)

Discussiones Mathematicae Graph Theory

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As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekulé structures (1-factors). A bipartite graph G is called elementary if G is connected and every edge belongs to a 1-factor of G. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given. A peripheral face f of a plane elementary graph is reducible, if the removal of the internal vertices and edges of the path that is the intersection...

Decompositions of quadrangle-free planar graphs

Oleg V. Borodin, Anna O. Ivanova, Alexandr V. Kostochka, Naeem N. Sheikh (2009)

Discussiones Mathematicae Graph Theory

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W. He et al. showed that a planar graph not containing 4-cycles can be decomposed into a forest and a graph with maximum degree at most 7. This degree restriction was improved to 6 by Borodin et al. We further lower this bound to 5 and show that it cannot be improved to 3.