Displaying similar documents to “Arc-transitive and s-regular Cayley graphs of valency five on Abelian groups”

Iterated neighborhood graphs

Martin Sonntag, Hanns-Martin Teichert (2012)

Discussiones Mathematicae Graph Theory

Similarity:

The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph ( V , E N ) where E N = a,b | a ≠ b, x,a ∈ E and x,b ∈ E for some x ∈ V. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite. We present some results concerning the k-iterated neighborhood graph N k ( G ) : = N ( N ( . . . N ( G ) ) ) of G. In particular we investigate conditions for G and k such that N k ( G ) becomes a complete graph.

Hexavalent ( G , s ) -transitive graphs

Song-Tao Guo, Xiao-Hui Hua, Yan-Tao Li (2013)

Czechoslovak Mathematical Journal

Similarity:

Let X be a finite simple undirected graph with a subgroup G of the full automorphism group Aut ( X ) . Then X is said to be ( G , s ) -transitive for a positive integer s , if G is transitive on s -arcs but not on ( s + 1 ) -arcs, and s -transitive if it is ( Aut ( X ) , s ) -transitive. Let G v be a stabilizer of a vertex v V ( X ) in G . Up to now, the structures of vertex stabilizers G v of cubic, tetravalent or pentavalent ( G , s ) -transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers G v of connected hexavalent...

On the minus domination number of graphs

Hailong Liu, Liang Sun (2004)

Czechoslovak Mathematical Journal

Similarity:

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...

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 ) N G [ u ] N G [ v ] , where N G [ x ] = N G ( x ) x . In the case where G is a claw-free graph, G* is equal to G². We define σ ° = m i n x 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.

A note on periodicity of the 2-distance operator

Bohdan Zelinka (2000)

Discussiones Mathematicae Graph Theory

Similarity:

The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class C f of all finite undirected graphs. If G is a graph from C f , then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.

Negative universality results for graphs

S.-D. Friedman, K. Thompson (2010)

Fundamenta Mathematicae

Similarity:

It is shown that in many forcing models there is no universal graph at the successors of regular cardinals. The proof, which is similar to the well-known proof for Cohen forcing, is extended to show that it is consistent to have no universal graph at the successor of a singular cardinal, and in particular at ω + 1 . Previously, little was known about universality at the successors of singulars. Analogous results show it is consistent not just that there is no single graph which embeds the...

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

Similarity:

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 ) n / 3 , and provide a characterization of graphs achieving this bound.

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 ( ) of gamma sets in the total graph on ℤₙ. We characterize the values of n for which I Γ ( ) is complete, bipartite, cycle, chordal and planar. Further, we prove that I Γ ( ) 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 ( ) .

4-cycle properties for characterizing rectagraphs and hypercubes

Khadra Bouanane, Abdelhafid Berrachedi (2017)

Czechoslovak Mathematical Journal

Similarity:

A ( 0 , 2 ) -graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of ( 0 , λ ) -graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free ( 0 , 2 ) -graph. ( 0 , 2 ) -graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in ( 0 , λ ) -graphs and more specifically...

Signed total domination number of a graph

Bohdan Zelinka (2001)

Czechoslovak Mathematical Journal

Similarity:

The signed total domination number of a graph is a certain variant of the domination number. If v is a vertex of a graph G , then N ( v ) is its oper neighbourhood, i.e. the set of all vertices adjacent to v in G . A mapping f : V ( G ) { - 1 , 1 } , where V ( G ) is the vertex set of G , is called a signed total dominating function (STDF) on G , if x N ( v ) f ( x ) 1 for each v V ( G ) . The minimum of values x V ( G ) f ( x ) , taken over all STDF’s of G , is called the signed total domination number of G and denoted by γ s t ( G ) . A theorem stating lower bounds for γ s t ( G ) is...

Potentially H-bigraphic sequences

Michael Ferrara, Michael Jacobson, John Schmitt, Mark Siggers (2009)

Discussiones Mathematicae Graph Theory

Similarity:

We extend the notion of a potentially H-graphic sequence as follows. Let A and B be nonnegative integer sequences. The sequence pair S = (A,B) is said to be bigraphic if there is some bipartite graph G = (X ∪ Y,E) such that A and B are the degrees of the vertices in X and Y, respectively. If S is a bigraphic pair, let σ(S) denote the sum of the terms in A. Given a bigraphic pair S, and a fixed bipartite graph H, we say that S is potentially H-bigraphic if there is some realization of...

On light subgraphs in plane graphs of minimum degree five

Stanislav Jendrol', Tomáš Madaras (1996)

Discussiones Mathematicae Graph Theory

Similarity:

A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is well known that a plane graph of minimum degree five contains light edges and light triangles. In this paper we show that every plane graph of minimum degree five contains also light stars K 1 , 3 and K 1 , 4 and a light 4-path P₄. The results obtained for K 1 , 3 and P₄ are best possible.

Decomposition of complete bipartite digraphs and even complete bipartite multigraphs into closed trails

Sylwia Cichacz (2007)

Discussiones Mathematicae Graph Theory

Similarity:

It has been shown [3] that any bipartite graph K a , b , where a, b are even integers, can be decomposed into closed trails with prescribed even lengths. In this article, we consider the corresponding question for directed bipartite graphs. We show that a complete directed bipartite graph K a , b is decomposable into directed closed trails of even lengths greater than 2, whenever these lengths sum up to the size of the digraph. We use this result to prove that complete bipartite multigraphs can be...