Extremal problems for forbidden pairs that imply hamiltonicity

Ralph Faudree; András Gyárfás

Displaying similar documents to “Extremal problems for forbidden pairs that imply hamiltonicity”

Measures of traceability in graphs

Varaporn Saenpholphat, Futaba Okamoto, Ping Zhang (2006)

Mathematica Bohemica

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For a connected graph $G$ of order $n \geq 3$ and an ordering $s v_{1}$ , $v_{2}, \dots, v_{n}$ of the vertices of $G$ , $d (s) = \sum_{i = 1}^{n - 1} d (v_{i}, v_{i + 1})$ , where $d (v_{i}, v_{i + 1})$ is the distance between $v_{i}$ and $v_{i + 1}$ . The traceable number $t (G)$ of $G$ is defined by $t (G) = min \}d (s)\{,$ where the minimum is taken over all sequences $s$ of the elements of $V (G)$ . It is shown that if $G$ is a nontrivial connected graph of order $n$ such that $l$ is the length of a longest path in $G$ and $p$ is the maximum size of a spanning linear forest in $G$ , then $2 n - 2 - p \leq t (G) \leq 2 n - 2 - l$ and both these bounds are sharp. We establish a formula for the traceable...

Variations on a sufficient condition for Hamiltonian graphs

Ahmed Ainouche, Serge Lapiquonne (2007)

Discussiones Mathematicae Graph Theory

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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...

Extension of several sufficient conditions for Hamiltonian graphs

Ahmed Ainouche (2006)

Discussiones Mathematicae Graph Theory

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Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖u)|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph $(K_{r} \cup K ₛ \cup K ₜ) \lor K ₂$ , 1 ≤ r ≤ s ≤ t or the graph $((n + 1) / 2) K ₁ \lor K_{(n - 1) / 2}$ . It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity...

A note on a new condition implying pancyclism

Evelyne Flandrin, Hao Li, Antoni Marczyk, Mariusz Woźniak (2001)

Discussiones Mathematicae Graph Theory

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We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to $K_{n / 2, n / 2}$ .

On integral sum graphs with a saturated vertex

Zhibo Chen (2010)

Czechoslovak Mathematical Journal

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As introduced by F. Harary in 1994, a graph $G$ is said to be an $i n t e g r a l$ $s u m$ $g r a p h$ if its vertices can be given a labeling $f$ with distinct integers so that for any two distinct vertices $u$ and $v$ of $G$ , $u v$ is an edge of $G$ if and only if $f (u) + f (v) = f (w)$ for some vertex $w$ in $G$ . We prove that every integral sum graph with a saturated vertex, except the complete graph $K_{3}$ , has edge-chromatic number equal to its maximum degree. (A vertex of a graph $G$ is said to be if it is adjacent to every...

Join of two graphs admits a nowhere-zero $3$ -flow

Saieed Akbari, Maryam Aliakbarpour, Naryam Ghanbari, Emisa Nategh, Hossein Shahmohamad (2014)

Czechoslovak Mathematical Journal

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Let $G$ be a graph, and $λ$ the smallest integer for which $G$ has a nowhere-zero $λ$ -flow, i.e., an integer $λ$ for which $G$ admits a nowhere-zero $λ$ -flow, but it does not admit a $(λ - 1)$ -flow. We denote the minimum flow number of $G$ by $Λ (G)$ . In this paper we show that if $G$ and $H$ are two arbitrary graphs and $G$ has no isolated vertex, then $Λ (G \lor H) \leq 3$ except two cases: (i) One of the graphs $G$ and $H$ is $K_{2}$ and the other is $1$ -regular. (ii) $H = K_{1}$ and $G$ is a graph with at least one isolated vertex or a component whose every...

Histories in path graphs

Ludovít Niepel (2007)

Discussiones Mathematicae Graph Theory

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For a given graph G and a positive integer r the r-path graph, $P_{r} (G)$ , has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r-1, and their union forms either a cycle or a path of length k+1 in G. Let $P_{r}^{k} (G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_{r}^{k} (G)$ . The k-history $P_{r}^{- k} (H)$ is a subgraph of G that is induced by all edges that take part in the recursive...

Restrained domination in unicyclic graphs

Johannes H. Hattingh, Ernst J. Joubert, Marc Loizeaux, Andrew R. Plummer, Lucas van der Merwe (2009)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted by $γ_{r} (G)$ , is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then $γ_{r} (U) \geq ⎡ n / 3 ⎤$ , and provide a characterization of graphs achieving this bound.

New edge neighborhood graphs

Ali A. Ali, Salar Y. Alsardary (1997)

Czechoslovak Mathematical Journal

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Let $G$ be an undirected simple connected graph, and $e = u v$ be an edge of $G$ . Let $N_{G} (e)$ be the subgraph of $G$ induced by the set of all vertices of $G$ which are not incident to $e$ but are adjacent to $u$ or $v$ . Let $𝒩_{e}$ be the class of all graphs $H$ such that, for some graph $G$ , $N_{G} (e) ≅ H$ for every edge $e$ of $G$ . Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in $𝒩_{e}$ . Balasubramanian and Alsardary [1] obtained some other graphs in $𝒩_{e}$ . In this paper we given some new graphs in $𝒩_{e}$ .

On well-covered graphs of odd girth 7 or greater

Bert Randerath, Preben Dahl Vestergaard (2002)

Discussiones Mathematicae Graph Theory

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A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth...

Factors and circuits in $K_{1, 3}$ -free graphs

Zdeněk Ryjáček (1989)

Banach Center Publications

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