Finding vertex-disjoint cycle cover of undirected graph using the least-squares method

Lamač, Jan; Vlasák, Miloslav

Displaying similar documents to “Finding vertex-disjoint cycle cover of undirected graph using the least-squares method”

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

Similarity:

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.

Cycle-pancyclism in bipartite tournaments II

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

Similarity:

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...

Cycle-pancyclism in bipartite tournaments I

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

Similarity:

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 <...

Intersection graph of gamma sets in the total graph

T. Tamizh Chelvam, T. Asir (2012)

Discussiones Mathematicae Graph Theory

Similarity:

In this paper, we consider the intersection graph $I_{Γ} (ℤ ₙ)$ of gamma sets in the total graph on ℤₙ. We characterize the values of n for which $I_{Γ} (ℤ ₙ)$ is complete, bipartite, cycle, chordal and planar. Further, we prove that $I_{Γ} (ℤ ₙ)$ is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of $I_{Γ} (ℤ ₙ)$ .

Chvátal's Condition cannot hold for both a graph and its complement

Alexandr V. Kostochka, Douglas B. West (2006)

Discussiones Mathematicae Graph Theory

Similarity:

Chvátal’s Condition is a sufficient condition for a spanning cycle in an n-vertex graph. The condition is that when the vertex degrees are d₁, ...,dₙ in nondecreasing order, i < n/2 implies that $d_{i} > i$ or $d_{n - i} \geq n - i$ . We prove that this condition cannot hold in both a graph and its complement, and we raise the problem of finding its asymptotic probability in the random graph with edge probability 1/2.

Variations on a sufficient condition for Hamiltonian graphs

Ahmed Ainouche, Serge Lapiquonne (2007)

Discussiones Mathematicae Graph Theory

Similarity:

Given a 2-connected graph G on n vertices, let G* be its partially square graph, obtained by adding edges uv whenever the vertices u,v have a common neighbor x satisfying the condition $N_{G} (x) \subseteq N_{G} [u] \cup N_{G} [v]$ , where $N_{G} [x] = N_{G} (x) \cup x$ . In particular, this condition is satisfied if x does not center a claw (an induced $K_{1, 3}$ ). Clearly G ⊆ G* ⊆ G², where G² is the square of G. For any independent triple X = x,y,z we define σ̅(X) = d(x) + d(y) + d(z) - |N(x) ∩ N(y) ∩ N(z)|. Flandrin et al. proved that a 2-connected graph G is hamiltonian...

Minimal and minimum size latin bitrades of each genus

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

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...

On the order of certain close to regular graphs without a matching of given size

Sabine Klinkenberg, Lutz Volkmann (2007)

Czechoslovak Mathematical Journal

Similarity:

A graph $G$ is a ${d, d + k}$ -graph, if one vertex has degree $d + k$ and the remaining vertices of $G$ have degree $d$ . In the special case of $k = 0$ , the graph $G$ is $d$ -regular. Let $k, p \geq 0$ and $d, n \geq 1$ be integers such that $n$ and $p$ are of the same parity. If $G$ is a connected ${d, d + k}$ -graph of order $n$ without a matching $M$ of size $2 | M | = n - p$ , then we show in this paper the following: If $d = 2$ , then $k \geq 2 (p + 2)$ and (i) $n \geq k + p + 6$ . If $d \geq 3$ is odd and $t$ an integer with $1 \leq t \leq p + 2$ , then (ii) $n \geq d + k + 1$ for $k \geq d (p + 2)$ , (iii) $n \geq d (p + 3) + 2 t + 1$ for $d (p + 2 - t) + t \leq k \leq d (p + 3 - t) + t - 3$ , (iv) $n \geq d (p + 3) + 2 p + 7$ for $k \leq p$ . If $d \geq 4$ is even, then (v) $n \geq d + k + 2 - η$ for $k \geq d (p + 3) + p + 4 + η$ , (vi) $n \geq d + k + p + 2 - 2 t = d (p + 4) + p + 6$ for $k = d (p + 3) + 4 + 2 t$ and $p \geq 1$ ,...

Remarks on partially square graphs, hamiltonicity and circumference

Hamamache Kheddouci (2001)

Discussiones Mathematicae Graph Theory

Similarity:

Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition $N_{G} (x) \subseteq N_{G} [u] \cup N_{G} [v]$ , where $N_{G} [x] = N_{G} (x) \cup x$ . In the case where G is a claw-free graph, G* is equal to G². We define $σ ° ₜ = m i n \sum_{x \in S} d_{G} (x) : S i s a n i n d e p e n d e n t s e t i n G * a n d | S | = t$ . We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.

On-line ranking number for cycles and paths

Erik Bruoth, Mirko Horňák (1999)

Discussiones Mathematicae Graph Theory

Similarity:

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....

Hamiltonicity of cubic Cayley graphs

Henry Glover, Dragan Marušič (2007)

Journal of the European Mathematical Society

Similarity:

Following a problem posed by Lovász in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a $(2, s, 3)$ -presentation, that is, for groups $G = 〈 a, b ∣ a^{2} = 1, b^{s} = 1, {(a b)}^{3} = 1, \dots 〉$ generated by an involution $a$ and an element $b$ of order $s \geq 3$ such that their product $a b$ has order $3$ . More precisely, it is shown that the Cayley graph $X = Cay (G, {a, b, b^{- 1}})$ has a Hamilton cycle when $| G |$ (and thus $s$ ) is congruent to 2 modulo 4, and has a...

On a family of cubic graphs containing the flower snarks

Jean-Luc Fouquet, Henri Thuillier, Jean-Marie Vanherpe (2010)

Discussiones Mathematicae Graph Theory

Similarity:

We consider cubic graphs formed with k ≥ 2 disjoint claws $C_{i} K_{1, 3}$ (0 ≤ i ≤ k-1) such that for every integer i modulo k the three vertices of degree 1 of $C_{i}$ are joined to the three vertices of degree 1 of $C_{i - 1}$ and joined to the three vertices of degree 1 of $C_{i + 1}$ . Denote by $t_{i}$ the vertex of degree 3 of $C_{i}$ and by T the set $t ₁, t ₂, . . ., t_{k - 1}$ . In such a way we construct three distinct graphs, namely FS(1,k), FS(2,k) and FS(3,k). The graph FS(j,k) (j ∈ 1,2,3) is the graph where the set of vertices $⋃_{i = 0}^{i = k - 1} V (C_{i}) ∖ T$ induce j cycles (note...

On short cycles in triangle-free oriented graphs

Yurong Ji, Shufei Wu, Hui Song (2018)

Czechoslovak Mathematical Journal

Similarity:

An orientation of a simple graph is referred to as an oriented graph. Caccetta and Häggkvist conjectured that any digraph on $n$ vertices with minimum outdegree $d$ contains a directed cycle of length at most $⌈ n / d ⌉$ . In this paper, we consider short cycles in oriented graphs without directed triangles. Suppose that $α_{0}$ is the smallest real such that every $n$ -vertex digraph with minimum outdegree at least $α_{0} n$ contains a directed triangle. Let $ϵ < (3 - 2 α_{0}) / (4 - 2 α_{0})$ be a positive real. We show that if $D$ is an oriented graph...