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Contractible edges in some k -connected graphs

Yingqiu YangLiang Sun — 2012

Czechoslovak Mathematical Journal

An edge e of a k -connected graph G is said to be k -contractible (or simply contractible) if the graph obtained from G by contracting e (i.e., deleting e and identifying its ends, finally, replacing each of the resulting pairs of double edges by a single edge) is still k -connected. In 2002, Kawarabayashi proved that for any odd integer k 5 , if G is a k -connected graph and G contains no subgraph D = K 1 + ( K 2 K 1 , 2 ) , then G has a k -contractible edge. In this paper, by generalizing this result, we prove that for any integer...

On the minus domination number of graphs

Hailong LiuLiang Sun — 2004

Czechoslovak Mathematical Journal

Let G = ( V , E ) be a simple graph. A 3 -valued function f V ( G ) { - 1 , 0 , 1 } is said to be a minus dominating function if for every vertex v V , f ( N [ v ] ) = u N [ v ] f ( u ) 1 , where N [ v ] is the closed neighborhood of v . The weight of a minus dominating function f on G is f ( V ) = v V f ( v ) . The minus domination number of a graph G , denoted by γ - ( G ) , equals the minimum weight of a minus dominating function on G . In this paper, the following two results are obtained. (1) If G is a bipartite graph of order n , then γ - ( G ) 4 n + 1 - 1 - n . (2) For any negative integer k and any positive integer m 3 , there exists...

On domination number of 4-regular graphs

Hailong LiuLiang Sun — 2004

Czechoslovak Mathematical Journal

Let G be a simple graph. A subset S V is a dominating set of G , if for any vertex v V - S there exists a vertex u S such that u v E ( G ) . The domination number, denoted by γ ( G ) , is the minimum cardinality of a dominating set. In this paper we prove that if G is a 4-regular graph with order n , then γ ( G ) 4 11 n .

Connected domination critical graphs with respect to relative complements

Xue-Gang ChenLiang Sun — 2006

Czechoslovak Mathematical Journal

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G . The minimum number of vertices in a connected dominating set of G is called the connected domination number of G , and is denoted by γ c ( G ) . Let G be a spanning subgraph of K s , s and let H be the complement of G relative to K s , s ; that is, K s , s = G H is a factorization of K s , s . The graph G is k - γ c -critical relative to K s , s if γ c ( G ) = k and γ c ( G + e ) < k for each edge e E ( H ) . First, we discuss some classes of graphs whether they are γ c -critical relative...

On total restrained domination in graphs

De-xiang MaXue-Gang ChenLiang Sun — 2005

Czechoslovak Mathematical Journal

In this paper we initiate the study of total restrained domination in graphs. Let G = ( V , E ) be a graph. A total restrained dominating set is a set S V where every vertex in V - S is adjacent to a vertex in S as well as to another vertex in V - S , and every vertex in S is adjacent to another vertex in S . The total restrained domination number of G , denoted by γ r t ( G ) , is the smallest cardinality of a total restrained dominating set of G . First, some exact values and sharp bounds for γ r t ( G ) are given in Section 2. Then the Nordhaus-Gaddum-type...

± sign pattern matrices that allow orthogonality

Yan Ling ShaoLiang SunYubin Gao — 2006

Czechoslovak Mathematical Journal

A sign pattern A is a ± sign pattern if A has no zero entries. A allows orthogonality if there exists a real orthogonal matrix B whose sign pattern equals A . Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for ± sign patterns with n - 1 N - ( A ) n + 1 to allow orthogonality.

An upper bound for domination number of 5-regular graphs

Hua Ming XingLiang SunXue-Gang Chen — 2006

Czechoslovak Mathematical Journal

Let G = ( V , E ) be a simple graph. A subset S V is a dominating set of G , if for any vertex u V - S , there exists a vertex v S such that u v E . The domination number, denoted by γ ( G ) , is the minimum cardinality of a dominating set. In this paper we will prove that if G is a 5-regular graph, then γ ( G ) 5 14 n .

On signed majority total domination in graphs

Hua Ming XingLiang SunXue-Gang Chen — 2005

Czechoslovak Mathematical Journal

We initiate the study of signed majority total domination in graphs. Let G = ( V , E ) be a simple graph. For any real valued function f V and S V , let f ( S ) = v S f ( v ) . A signed majority total dominating function is a function f V { - 1 , 1 } such that f ( N ( v ) ) 1 for at least a half of the vertices v V . The signed majority total domination number of a graph G is γ m a j t ( G ) = min { f ( V ) f is a signed majority total dominating function on G } . We research some properties of the signed majority total domination number of a graph G and obtain a few lower bounds of γ m a j t ( G ) .

On signed distance- k -domination in graphs

Hua Ming XingLiang SunXue-Gang Chen — 2006

Czechoslovak Mathematical Journal

The signed distance- k -domination number of a graph is a certain variant of the signed domination number. If v is a vertex of a graph G , the open k -neighborhood of v , denoted by N k ( v ) , is the set N k ( v ) = { u u v and d ( u , v ) k } . N k [ v ] = N k ( v ) { v } is the closed k -neighborhood of v . A function f V { - 1 , 1 } is a signed distance- k -dominating function of G , if for every vertex v V , f ( N k [ v ] ) = u N k [ v ] f ( u ) 1 . The signed distance- k -domination number, denoted by γ k , s ( G ) , is the minimum weight of a signed distance- k -dominating function on G . The values of γ 2 , s ( G ) are found for graphs with small diameter,...

Multiplicity solutions of a class fractional Schrödinger equations

Li-Jiang JiaBin GeYing-Xin CuiLiang-Liang Sun — 2017

Open Mathematics

In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations (−Δ)su+V(x)u=λf(x,u)inRN, ( - Δ ) s u + V ( x ) u = λ f ( x , u ) in N , where [...] (−Δ)su(x)=2limε→0∫RN∖Bε(X)u(x)−u(y)|x−y|N+2sdy,x∈RN ( - Δ ) s u ( x ) = 2 lim ε 0 N B ε ( X ) u ( x ) - u ( y ) | x - y | N + 2 s d y , x N is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.

A note on the independent domination number of subset graph

Xue-Gang ChenDe-xiang MaHua Ming XingLiang Sun — 2005

Czechoslovak Mathematical Journal

The independent domination number i ( G ) (independent number β ( G ) ) is the minimum (maximum) cardinality among all maximal independent sets of G . Haviland (1995) conjectured that any connected regular graph G of order n and degree δ 1 2 n satisfies i ( G ) 2 n 3 δ 1 2 δ . For 1 k l m , the subset graph S m ( k , l ) is the bipartite graph whose vertices are the k - and l -subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i ( S m ( k , l ) ) and prove that...

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