Displaying similar documents to “A note on the independent domination number versus the domination number in bipartite graphs”

Edit distance measure for graphs

Tomasz Dzido, Krzysztof Krzywdziński (2015)

Czechoslovak Mathematical Journal

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In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for g ( n , l ) , the biggest number k guaranteeing that there exist l graphs on n vertices, each two having edit distance at least k . By edit distance of two graphs G , F we mean the number of edges needed to be added to or deleted from graph G to obtain graph F . This new extremal number g ( n , l ) is closely linked to the edit distance of graphs. Using probabilistic methods we show...

Remarks on D -integral complete multipartite graphs

Pavel Híc, Milan Pokorný (2016)

Czechoslovak Mathematical Journal

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A graph is called distance integral (or D -integral) if all eigenvalues of its distance matrix are integers. In their study of D -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on D -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs K p 1 , p 2 , p 3 with p 1 < p 2 < p 3 , and K p 1 , p 2 , p 3 , p 4 with p 1 < p 2 < p 3 < p 4 , as well as the infinite classes of distance integral...

Intrinsic linking and knotting are arbitrarily complex

Erica Flapan, Blake Mellor, Ramin Naimi (2008)

Fundamenta Mathematicae

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We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, | l k ( Q i , Q j ) | α and | a ( Q i ) | α , where a ( Q i ) denotes the second coefficient of the Conway polynomial of Q i .

On distinguishing and distinguishing chromatic numbers of hypercubes

Werner Klöckl (2008)

Discussiones Mathematicae Graph Theory

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The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number χ D ( G ) of G. Extending these concepts to infinite graphs we prove that D ( Q ) = 2 and χ D ( Q ) = 3 , where Q denotes the hypercube of countable dimension. We also show that χ D ( Q ) = 4 , thereby completing the investigation of finite hypercubes with respect to χ D . Our...

Edge-colouring of graphs and hereditary graph properties

Samantha Dorfling, Tomáš Vetrík (2016)

Czechoslovak Mathematical Journal

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Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph G , a hereditary graph property 𝒫 and l 1 we define χ 𝒫 , l ' ( G ) to be the minimum number of colours needed to properly colour the edges of G , such that any subgraph of G induced by edges coloured by (at most) l colours is in 𝒫 . We present a necessary and sufficient condition for the existence of χ 𝒫 , l ' ( G ) . We focus on edge-colourings of graphs with respect to the hereditary...

The real symmetric matrices of odd order with a P-set of maximum size

Zhibin Du, Carlos M. da Fonseca (2016)

Czechoslovak Mathematical Journal

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Suppose that A is a real symmetric matrix of order n . Denote by m A ( 0 ) the nullity of A . For a nonempty subset α of { 1 , 2 , ... , n } , let A ( α ) be the principal submatrix of A obtained from A by deleting the rows and columns indexed by α . When m A ( α ) ( 0 ) = m A ( 0 ) + | α | , we call α a P-set of A . It is known that every P-set of A contains at most n / 2 elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As...

Embedding products of graphs into Euclidean spaces

Mikhail Skopenkov (2003)

Fundamenta Mathematicae

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For any collection of graphs G , . . . , G N we find the minimal dimension d such that the product G × . . . × G N is embeddable into d (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and ( K 3 , 3 ) are not embeddable into 2 n , where K₅ and K 3 , 3 are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding L k O S 2 n - 1 , where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

Generalized 3-edge-connectivity of Cartesian product graphs

Yuefang Sun (2015)

Czechoslovak Mathematical Journal

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The generalized k -connectivity κ k ( G ) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k -edge-connectivity which is defined as λ k ( G ) = min { λ ( S ) : S V ( G ) and | S | = k } , where λ ( S ) denotes the maximum number of pairwise edge-disjoint trees T 1 , T 2 , ... , T in G such that S V ( T i ) for 1 i . In this paper we prove that for any two connected graphs G and H we have λ 3 ( G H ) λ 3 ( G ) + λ 3 ( H ) , where G H is the Cartesian product of G and H . Moreover, the bound is sharp. We also...

On γ-labelings of trees

Gary Chartrand, David Erwin, Donald W. VanderJagt, Ping Zhang (2005)

Discussiones Mathematicae Graph Theory

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Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → 0,1,2,...,m that induces a labeling f’: E(G) → 1,2,...,m of the edges of G defined by f’(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is v a l ( f ) = Σ e E ( G ) f ' K ( e ) . The maximum value of a γ-labeling of G is defined as v a l m a x ( G ) = m a x v a l ( f ) : f i s a γ - l a b e l i n g o f G ; while the minimum value of a γ-labeling of G is v a l m i n ( G ) = m i n v a l ( f ) : f i s a γ - l a b e l i n g o f G ; The values v a l m a x ( S p , q ) and v a l m i n ( S p , q ) are determined for double stars S p , q . We present characterizations of connected graphs G of order n for which...

The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths

Halina Bielak, Kinga Dąbrowska (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The Ramsey number R ( G , H ) for a pair of graphs G and H is defined as the smallest integer n such that, for any graph F on n vertices, either F contains G or F ¯ contains H as a subgraph, where F ¯ denotes the complement of F . We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers R ( K 1 + L n , P m ) and R ( K 1 + L n , C m ) for some integers m , n , where L n is...

Persistency in the Traveling Salesman Problem on Halin graphs

Vladimír Lacko (2000)

Discussiones Mathematicae Graph Theory

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For the Traveling Salesman Problem (TSP) on Halin graphs with three types of cost functions: sum, bottleneck and balanced and with arbitrary real edge costs we compute in polynomial time the persistency partition E A l l , E S o m e , E N o n e of the edge set E, where: E A l l = e ∈ E, e belongs to all optimum solutions, E N o n e = e ∈ E, e does not belong to any optimum solution and E S o m e = e ∈ E, e belongs to some but not to all optimum solutions.

On 𝓕-independence in graphs

Frank Göring, Jochen Harant, Dieter Rautenbach, Ingo Schiermeyer (2009)

Discussiones Mathematicae Graph Theory

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Let be a set of graphs and for a graph G let α ( G ) and α * ( G ) denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on α ( G ) and α * ( G ) are presented.

On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

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Let G be a finite group. The intersection graph Δ G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G , and two distinct vertices X and Y are adjacent if X Y 1 , where 1 denotes the trivial subgroup of order 1 . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters...

Some remarks on α-domination

Franz Dahme, Dieter Rautenbach, Lutz Volkmann (2004)

Discussiones Mathematicae Graph Theory

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Let α ∈ (0,1) and let G = ( V G , E G ) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set D V G is called an α-dominating set of G, if | N G ( u ) D | α d G ( u ) for all u V G D . We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.

The Turán number of the graph 3 P 4

Halina Bielak, Sebastian Kieliszek (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let e x ( n , G ) denote the maximum number of edges in a graph on n vertices which does not contain G as a subgraph. Let P i denote a path consisting of i vertices and let m P i denote m disjoint copies of P i . In this paper we count e x ( n , 3 P 4 ) .

On path-quasar Ramsey numbers

Binlong Li, Bo Ning (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let G 1 and G 2 be two given graphs. The Ramsey number R ( G 1 , G 2 ) is the least integer r such that for every graph G on r vertices, either G contains a G 1 or G ¯ contains a G 2 . Parsons gave a recursive formula to determine the values of R ( P n , K 1 , m ) , where P n is a path on n vertices and K 1 , m is a star on m + 1 vertices. In this note, we study the Ramsey numbers R ( P n , K 1 F m ) , where F m is a linear forest on m vertices. We determine the exact values of R ( P n , K 1 F m ) for the cases m n and m 2 n , and for the case that F m has no odd component. Moreover, we...