Displaying similar documents to “Some results on semi-total signed graphs”

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...

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...

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 ) .

Classification of rings with toroidal Jacobson graph

Krishnan Selvakumar, Manoharan Subajini (2016)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with nonzero identity and J ( R ) the Jacobson radical of R . The Jacobson graph of R , denoted by 𝔍 R , is defined as the graph with vertex set R J ( R ) such that two distinct vertices x and y are adjacent if and only if 1 - x y is not a unit of R . The genus of a simple graph G is the smallest nonnegative integer n such that G can be embedded into an orientable surface S n . In this paper, we investigate the genus number of the compact Riemann surface in which 𝔍 R can be embedded and...

On the bounds of Laplacian eigenvalues of k -connected graphs

Xiaodan Chen, Yaoping Hou (2015)

Czechoslovak Mathematical Journal

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Let μ n - 1 ( G ) be the algebraic connectivity, and let μ 1 ( G ) be the Laplacian spectral radius of a k -connected graph G with n vertices and m edges. In this paper, we prove that μ n - 1 ( G ) 2 n k 2 ( n ( n - 1 ) - 2 m ) ( n + k - 2 ) + 2 k 2 , with equality if and only if G is the complete graph K n or K n - e . Moreover, if G is non-regular, then μ 1 ( G ) < 2 Δ - 2 ( n Δ - 2 m ) k 2 2 ( n Δ - 2 m ) ( n 2 - 2 n + 2 k ) + n k 2 , where Δ stands for the maximum degree of G . Remark that in some cases, these two inequalities improve some previously known results.

Complete pairs of coanalytic sets

Jean Saint Raymond (2007)

Fundamenta Mathematicae

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Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of ω ω there exists a continuous function f : ω ω X such that f - 1 ( C ) = D and f - 1 ( C ) = D . We give several explicit examples of complete pairs of coanalytic sets.

Nonempty intersection of longest paths in a graph with a small matching number

Fuyuan Chen (2015)

Czechoslovak Mathematical Journal

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A maximum matching of a graph G is a matching of G with the largest number of edges. The matching number of a graph G , denoted by α ' ( G ) , is the number of edges in a maximum matching of G . In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai’s conjecture...

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 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 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...

Size of the giant component in a random geometric graph

Ghurumuruhan Ganesan (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper, we study the size of the giant component C G in the random geometric graph G = G ( n , r n , f ) of n nodes independently distributed each according to a certain density f ( · ) in [ 0 , 1 ] 2 satisfying inf x [ 0 , 1 ] 2 f ( x ) g t ; 0 . If c 1 n r n 2 c 2 log n n for some positive constants c 1 , c 2 and n r n 2 as n , we show that the giant component of G contains at least n - o ( n ) nodes with probability at least 1 - e - β n r n 2 for all n and for some positive constant β . We also obtain estimates on the diameter and number of the non-giant components of G .