Displaying similar documents to “Characterizations of planar plick 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...

Radio Graceful Hamming Graphs

Amanda Niedzialomski (2016)

Discussiones Mathematicae Graph Theory

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For k ∈ ℤ+ and G a simple, connected graph, a k-radio labeling f : V (G) → ℤ+ of G requires all pairs of distinct vertices u and v to satisfy |f(u) − f(v)| ≥ k + 1 − d(u, v). We consider k-radio labelings of G when k = diam(G). In this setting, f is injective; if f is also surjective onto {1, 2, . . . , |V (G)|}, then f is a consecutive radio labeling. Graphs that can be labeled with such a labeling are called radio graceful. In this paper, we give two results on the existence of radio...

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.

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...

Note on enumeration of labeled split graphs

Vladislav Bína, Jiří Přibil (2015)

Commentationes Mathematicae Universitatis Carolinae

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The paper brings explicit formula for enumeration of vertex-labeled split graphs with given number of vertices. The authors derive this formula combinatorially using an auxiliary assertion concerning number of split graphs with given clique number. In conclusion authors discuss enumeration of vertex-labeled bipartite graphs, i.e., a graphical class defined in a similar manner to the class of split graphs.

The Thickness of Amalgamations and Cartesian Product of Graphs

Yan Yang, Yichao Chen (2017)

Discussiones Mathematicae Graph Theory

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The thickness of a graph is the minimum number of planar spanning subgraphs into which the graph can be decomposed. It is a measurement of the closeness to the planarity of a graph, and it also has important applications to VLSI design, but it has been known for only few graphs. We obtain the thickness of vertex-amalgamation and bar-amalgamation of graphs, the lower and upper bounds for the thickness of edge-amalgamation and 2-vertex-amalgamation of graphs, respectively. We also study...

Characterizing Cartesian fixers and multipliers

Stephen Benecke, Christina M. Mynhardt (2012)

Discussiones Mathematicae Graph Theory

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Let G ☐ H denote the Cartesian product of the graphs G and H. In 2004, Hartnell and Rall [On dominating the Cartesian product of a graph and K₂, Discuss. Math. Graph Theory 24(3) (2004), 389-402] characterized prism fixers, i.e., graphs G for which γ(G ☐ K₂) = γ(G), and noted that γ(G ☐ Kₙ) ≥ min{|V(G)|, γ(G)+n-2}. We call a graph G a consistent fixer if γ(G ☐ Kₙ) = γ(G)+n-2 for each n such that 2 ≤ n < |V(G)|- γ(G)+2, and characterize this class of graphs. Also...

On dually compact closed classes of graphs and BFS-constructible graphs

Norbert Polat (2003)

Discussiones Mathematicae Graph Theory

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A class C of graphs is said to be dually compact closed if, for every infinite G ∈ C, each finite subgraph of G is contained in a finite induced subgraph of G which belongs to C. The class of trees and more generally the one of chordal graphs are dually compact closed. One of the main part of this paper is to settle a question of Hahn, Sands, Sauer and Woodrow by showing that the class of bridged graphs is dually compact closed. To prove this result we use the concept of constructible...

The crossing numbers of join products of paths with graphs of order four

Marián Klešč, Stefan Schrötter (2011)

Discussiones Mathematicae Graph Theory

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Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87-97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numbers of graphs obtained as join product of two graphs. In the paper, the exact values of crossing...