On the structural result on normal plane maps

Tomás Madaras; Andrea Marcinová

Displaying similar documents to “On the structural result on normal plane maps”

How many clouds cover the plane?

James H. Schmerl (2003)

Fundamenta Mathematicae

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The plane can be covered by n + 2 clouds iff $2^{ℵ ₀} \leq ℵ ₙ$ .

On light subgraphs in plane graphs of minimum degree five

Stanislav Jendrol&#039;, Tomáš Madaras (1996)

Discussiones Mathematicae Graph Theory

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A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is well known that a plane graph of minimum degree five contains light edges and light triangles. In this paper we show that every plane graph of minimum degree five contains also light stars $K_{1, 3}$ and $K_{1, 4}$ and a light 4-path P₄. The results obtained for $K_{1, 3}$ and P₄ are best possible.

Domination and independence subdivision numbers of graphs

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi (2000)

Discussiones Mathematicae Graph Theory

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The domination subdivision number $s d_{γ} (G)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of...

On the adjacent eccentric distance sum of graphs

Halina Bielak, Katarzyna Wolska (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we show bounds for the adjacent eccentric distance sum of graphs in terms of Wiener index, maximum degree and minimum degree. We extend some earlier results of Hua and Yu [Bounds for the Adjacent Eccentric Distance Sum, International Mathematical Forum, Vol. 7 (2002) no. 26, 1289–1294]. The adjacent eccentric distance sum index of the graph $G$ is defined as $ξ^{s v} (G) = \sum_{v \in V (G)} \frac{ε (v) D (v)}{d e g (v)},$ where $ε (v)$ is the eccentricity of the vertex $v$ , $d e g (v)$ is the degree of the vertex $v$ and $D (v) = \sum_{u \in V (G)} d (u, v)$ is the sum of all distances from...

On locating-domination in graphs

Mustapha Chellali, Malika Mimouni, Peter J. Slater (2010)

Discussiones Mathematicae Graph Theory

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A set D of vertices in a graph G = (V,E) is a locating-dominating set (LDS) if for every two vertices u,v of V-D the sets N(u)∩ D and N(v)∩ D are non-empty and different. The locating-domination number $γ_{L} (G)$ is the minimum cardinality of a LDS of G, and the upper locating-domination number, $Γ_{L} (G)$ is the maximum cardinality of a minimal LDS of G. We present different bounds on $Γ_{L} (G)$ and $γ_{L} (G)$ .

Domination Subdivision Numbers

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi, David P. Jacobs, James Knisely, Lucas C. van der Merwe (2001)

Discussiones Mathematicae Graph Theory

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A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number $s d_{γ} (G)$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that $1 \leq s d_{γ} (G) \leq 3$ for any graph G. We give a counterexample to this conjecture. On the other hand,...

The Niemytzki plane is $ϰ$ -metrizable

Wojciech Bielas, Andrzej Kucharski, Szymon Plewik (2021)

Mathematica Bohemica

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We prove that the Niemytzki plane is $ϰ$ -metrizable and we try to explain the differences between the concepts of a stratifiable space and a $ϰ$ -metrizable space. Also, we give a characterisation of $ϰ$ -metrizable spaces which is modelled on the version described by Chigogidze.

On-line ranking number for cycles and paths

Erik Bruoth, Mirko Horňák (1999)

Discussiones Mathematicae Graph Theory

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A k-ranking of a graph G is a colouring φ:V(G) → 1,...,k such that any path in G with endvertices x,y fulfilling φ(x) = φ(y) contains an internal vertex z with φ(z) > φ(x). On-line ranking number $χ *_{r} (G)$ of a graph G is a minimum k such that G has a k-ranking constructed step by step if vertices of G are coming and coloured one by one in an arbitrary order; when colouring a vertex, only edges between already present vertices are known. Schiermeyer, Tuza and Voigt proved that $χ *_{r} (P ₙ) < 3 l o g ₂ n$ for n ≥ 2....

On subgraphs without large components

Glenn G. Chappell, John Gimbel (2017)

Mathematica Bohemica

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We consider, for a positive integer $k$ , induced subgraphs in which each component has order at most $k$ . Such a subgraph is said to be $k$ -divided. We show that finding large induced subgraphs with this property is NP-complete. We also consider a related graph-coloring problem: how many colors are required in a vertex coloring in which each color class induces a $k$ -divided subgraph. We show that the problem of determining whether some given number of colors suffice is NP-complete, even for...

A spectral bound for graph irregularity

Felix Goldberg (2015)

Czechoslovak Mathematical Journal

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The imbalance of an edge $e = {u, v}$ in a graph is defined as $i (e) = | d (u) - d (v) |$ , where $d (\cdot)$ is the vertex degree. The irregularity $I (G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$ . This concept was introduced by Albertson who proved that $I (G) \leq 4 n^{3} / 27$ (where $n = | V (G) |$ ) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves...

Stronger bounds for generalized degrees and Menger path systems

R.J. Faudree, Zs. Tuza (1995)

Discussiones Mathematicae Graph Theory

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For positive integers d and m, let $P_{d, m} (G)$ denote the property that between each pair of vertices of the graph G, there are m internally vertex disjoint paths of length at most d. For a positive integer t a graph G satisfies the minimum generalized degree condition δₜ(G) ≥ s if the cardinality of the union of the neighborhoods of each set of t vertices of G is at least s. Generalized degree conditions that ensure that $P_{d, m} (G)$ is satisfied have been investigated. In particular, it has been shown,...

On double domination in graphs

Jochen Harant, Michael A. Henning (2005)

Discussiones Mathematicae Graph Theory

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In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number $γ_{\times 2} (G)$ . A function f(p) is defined, and it is shown that $γ_{\times 2} (G) = m i n f (p)$ , where the minimum is taken over the n-dimensional cube $C ⁿ = p = (p ₁, . . ., p ₙ) | p_{i} \in I R, 0 \leq p_{i} \leq 1, i = 1, . . ., n$ . Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then $γ_{\times 2} (G) \leq ((l n (1 + d) + l n δ + 1) / δ) n$ .

Neighbor sum distinguishing list total coloring of IC-planar graphs without 5-cycles

Donghan Zhang (2022)

Czechoslovak Mathematical Journal

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Let $G = (V (G), E (G))$ be a simple graph and $E_{G} (v)$ denote the set of edges incident with a vertex $v$ . A neighbor sum distinguishing (NSD) total coloring $φ$ of $G$ is a proper total coloring of $G$ such that $\sum_{z \in E_{G} (u) \cup {u}} φ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} φ (z)$ for each edge $u v \in E (G)$ . Pilśniak and Woźniak asserted in 2015 that each graph with maximum degree $Δ$ admits an NSD total $(Δ + 3)$ -coloring. We prove that the list version of this conjecture holds for any IC-planar graph with $Δ \geq 11$ but without $5$ -cycles by applying the Combinatorial Nullstellensatz.

Note on improper coloring of $1$ -planar graphs

Yanan Chu, Lei Sun, Jun Yue (2019)

Czechoslovak Mathematical Journal

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A graph $G = (V, E)$ is called improperly $(d_{1}, \dots, d_{k})$ -colorable if the vertex set $V$ can be partitioned into subsets $V_{1}, \dots, V_{k}$ such that the graph $G [V_{i}]$ induced by the vertices of $V_{i}$ has maximum degree at most $d_{i}$ for all $1 \leq i \leq k$ . In this paper, we mainly study the improper coloring of $1$ -planar graphs and show that $1$ -planar graphs with girth at least $7$ are $(2, 0, 0, 0)$ -colorable.

Roman bondage in graphs

Nader Jafari Rad, Lutz Volkmann (2011)

Discussiones Mathematicae Graph Theory

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A Roman dominating function on a graph G is a function f:V(G) → 0,1,2 satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value $f (V (G)) = \sum_{u \in V (G)} f (u)$ . The Roman domination number, $γ_{R} (G)$ , of G is the minimum weight of a Roman dominating function on G. In this paper, we define the Roman bondage $b_{R} (G)$ of a graph G with maximum degree at least two to be the minimum cardinality of all sets E’ ⊆ E(G)...

Color-bounded hypergraphs, V: host graphs and subdivisions

Csilla Bujtás, Zsolt Tuza, Vitaly Voloshin (2011)

Discussiones Mathematicae Graph Theory

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A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = E₁,...,Eₘ, together with integers $s_{i}$ and $t_{i}$ satisfying $1 \leq s_{i} \leq t_{i} \leq | E_{i} |$ for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge $E_{i}$ satisfies $s_{i} \leq | φ (E_{i}) | \leq t_{i}$ . The hypergraph ℋ is colorable if it admits at least one proper coloring. We consider hypergraphs ℋ over a “host graph”, that means a graph G on the same vertex set X as ℋ, such that each $E_{i}$ induces a connected subgraph in G....

Turán number of two vertex-disjoint copies of cliques

Caiyun Hu (2024)

Czechoslovak Mathematical Journal

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The Turán number of a given graph $H$ , denoted by $ex (n, H)$ , is the maximum number of edges in an $H$ -free graph on $n$ vertices. Applying a well-known result of Hajnal and Szemerédi, we determine the Turán number $ex (n, K_{p} \cup K_{q}$ ) of a vertex-disjoint union of cliques $K_{p}$ and $K_{q}$ for all values of $n$ .