Short cycles of low weight in normal plane maps with minimum degree 5

Oleg V. Borodin; Douglas R. Woodall

Displaying similar documents to “Short cycles of low weight in normal plane maps with minimum degree 5”

On the structural result on normal plane maps

Tomás Madaras, Andrea Marcinová (2002)

Discussiones Mathematicae Graph Theory

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We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree Δ(G) ≤ D, D ≥ 12 is proved to be upper bounded by $6 + [(2 D + 12) / (D - 2)] ({(D - 1)}^{(t - 1)} - 1)$ . This improves a recent bound $6 + [(3 D + 3) / (D - 2)] ({(D - 1)}^{t - 1} - 1)$ , D ≥ 8 by Jendrol’ and Skupień, and the upper bound for distance-2 chromatic number.

Pairs of forbidden class of subgraphs concerning $K_{1, 3}$ and P₆ to have a cycle containing specified vertices

Takeshi Sugiyama, Masao Tsugaki (2009)

Discussiones Mathematicae Graph Theory

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In [3], Faudree and Gould showed that if a 2-connected graph contains no $K_{1, 3}$ and P₆ as an induced subgraph, then the graph is hamiltonian. In this paper, we consider the extension of this result to cycles passing through specified vertices. We define the families of graphs which are extension of the forbidden pair $K_{1, 3}$ and P₆, and prove that the forbidden families implies the existence of cycles passing through specified vertices.

A note on arc-disjoint cycles in tournaments

Jan Florek (2014)

Colloquium Mathematicae

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We prove that every vertex v of a tournament T belongs to at least $m a x m i n δ ⁺ (T), 2 δ ⁺ (T) - d ⁺_{T} (v) + 1, m i n δ ¯ (T), 2 δ ¯ (T) - d ¯_{T} (v) + 1$ arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and $d ⁺_{T} (v)$ (or $d ¯_{T} (v)$ ) is the out-degree (resp. in-degree) of v.

The structure and existence of 2-factors in iterated line graphs

Michael Ferrara, Ronald J. Gould, Stephen G. Hartke (2007)

Discussiones Mathematicae Graph Theory

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We prove several results about the structure of 2-factors in iterated line graphs. Specifically, we give degree conditions on G that ensure L²(G) contains a 2-factor with every possible number of cycles, and we give a sufficient condition for the existence of a 2-factor in L²(G) with all cycle lengths specified. We also give a characterization of the graphs G where $L^{k} (G)$ contains a 2-factor.

Cycle-pancyclism in bipartite tournaments I

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ [(n+4)/2], there exists a directed cycle $C_{h (k)}$ of length h(k), h(k) ∈ k,k-2 with $| A (C_{h (k)}) \cap A (γ) | \geq h (k) - 3$ and the result is best possible. In a forthcoming paper the case of directed cycles of length k, k even and k <...

Independent cycles and paths in bipartite balanced graphs

Beata Orchel, A. Paweł Wojda (2008)

Discussiones Mathematicae Graph Theory

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Bipartite graphs G = (L,R;E) and H = (L’,R’;E’) are bi-placeabe if there is a bijection f:L∪R→ L’∪R’ such that f(L) = L’ and f(u)f(v) ∉ E’ for every edge uv ∈ E. We prove that if G and H are two bipartite balanced graphs of order |G| = |H| = 2p ≥ 4 such that the sizes of G and H satisfy ||G|| ≤ 2p-3 and ||H|| ≤ 2p-2, and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs. This result implies easily that...

Equations relating factors in decompositions into factors of some family of plane triangulations, and applications (with an appendix by Andrzej Schinzel)

Jan Florek (2015)

Colloquium Mathematicae

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Let be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in dual terms) that every graph P ∈ has a decomposition into factors P₀, P₁, P₂ (indexed by elements of the cyclic group Q = 0,1,2) such that every factor $P_{q}$ consists of two induced paths of the same length M(q), and K(q) - 1 induced cycles of the same length 2M(q). For q ∈ Q, we define an integer S⁺(q) such that the vector (K(q),M(q),S⁺(q)) determines the graph...

Cycle-pancyclism in bipartite tournaments II

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T[V(γ)]? It is proved that for an even k in the range (n+6)/2 ≤ k ≤ n-2, there exists a directed cycle $C_{h (k)}$ of length h(k), h(k) ∈ k,k-2 with $| A (C_{h (k)}) \cap A (γ) | \geq h (k) - 4$ and the result is best possible. In a previous paper a similar result for 4 ≤ k ≤ (n+4)/2 was...

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.

On ${(4, 1)}^{*}$ -choosability of toroidal graphs without chordal 7-cycles and adjacent 4-cycles

Haihui Zhang (2013)

Commentationes Mathematicae Universitatis Carolinae

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A graph $G$ is called ${(k, d)}^{*}$ -choosable if for every list assignment $L$ satisfying $| L (v) | = k$ for all $v \in V (G)$ , there is an $L$ -coloring of $G$ such that each vertex of $G$ has at most $d$ neighbors colored with the same color as itself. In this paper, it is proved that every toroidal graph without chordal 7-cycles and adjacent 4-cycles is ${(4, 1)}^{*}$ -choosable.

On local structure of 1-planar graphs of minimum degree 5 and girth 4

Dávid Hudák, Tomás Madaras (2009)

Discussiones Mathematicae Graph Theory

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A graph is 1-planar if it can be embedded in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree 5 and girth 4 contains (1) a 5-vertex adjacent to an ≤ 6-vertex, (2) a 4-cycle whose every vertex has degree at most 9, (3) a $K_{1, 4}$ with all vertices having degree at most 11.

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 arbitrarily vertex decomposable unicyclic graphs with dominating cycle

Sylwia Cichacz, Irmina A. Zioło (2006)

Discussiones Mathematicae Graph Theory

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A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,...,nₖ) of positive integers such that $\sum_{i = 1}^{k} n_{i} = n$ , there exists a partition (V₁,...,Vₖ) of vertex set of G such that for every i ∈ 1,...,k the set $V_{i}$ induces a connected subgraph of G on $n_{i}$ vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We also characterize all such graphs with at most four hanging vertices such that exactly two of them have a common neighbour. ...

An upper bound on the basis number of the powers of the complete graphs

Salar Y. Alsardary (2001)

Czechoslovak Mathematical Journal

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The basis number of a graph $G$ is defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$ -fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is $\leq 2$ . Schmeichel proved that the basis number of the complete graph $K_{n}$ is at most $3$ . We generalize the result of Schmeichel by showing that the basis number of the $d$ -th power of $K_{n}$ is at most $2 d + 1$ .

Minimal and minimum size latin bitrades of each genus

James Lefevre, Diane Donovan, Nicholas J. Cavenagh, Aleš Drápal (2007)

Commentationes Mathematicae Universitatis Carolinae

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Suppose that $T^{\circ}$ and $T^{☆}$ are partial latin squares of order $n$ , with the property that each row and each column of $T^{\circ}$ contains the same set of entries as the corresponding row or column of $T^{☆}$ . In addition, suppose that each cell in $T^{\circ}$ contains an entry if and only if the corresponding cell in $T^{☆}$ contains an entry, and these entries (if they exist) are different. Then the pair $T = (T^{\circ}, T^{☆})$ forms a . The of $T$ is the total number of filled cells in $T^{\circ}$ (equivalently $T^{☆}$ ). The latin bitrade is if there is no...