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Displaying 341 – 360 of 908

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On Lee's conjecture and some results

Lixia Fan, Zhihe Liang (2009)

Discussiones Mathematicae Graph Theory

S.M. Lee proposed the conjecture: for any n > 1 and any permutation f in S(n), the permutation graph P(Pₙ,f) is graceful. For any integer n > 1 and permutation f in S(n), we discuss the gracefulness of the permutation graph P(Pₙ,f) if f = k = 0 l - 1 ( m + 2 k , m + 2 k + 1 ) , and k = 0 l - 1 ( m + 4 k , m + 4 k + 2 ) ( m + 4 k + 1 , m + 4 k + 3 ) for any positive integers m and l.

On L-ideal-based L-zero-divisor graphs

S. Ebrahimi Atani, M. Shajari Kohan (2011)

Discussiones Mathematicae - General Algebra and Applications

In a manner analogous to a commutative ring, the L-ideal-based L-zero-divisor graph of a commutative ring R can be defined as the undirected graph Γ(μ) for some L-ideal μ of R. The basic properties and possible structures of the graph Γ(μ) are studied.

On light subgraphs in plane graphs of minimum degree five

Stanislav Jendrol', Tomáš Madaras (1996)

Discussiones Mathematicae Graph Theory

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.

On •-Line Signed Graphs L•(S)

Deepa Sinha, Ayushi Dhama (2015)

Discussiones Mathematicae Graph Theory

A signed graph (or sigraph for short) is an ordered pair S = (Su,σ), where Su is a graph, G = (V,E), called the underlying graph of S and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a sigraph S its •-line sigraph, L•(S) is the sigraph in which the edges of S are represented as vertices, two of these vertices are defined adjacent whenever the corresponding edges in S have a vertex in common, any such L-edge ee′ has the sign given by the product of the signs of the...

On local structure of 1-planar graphs of minimum degree 5 and girth 4

Dávid Hudák, Tomás Madaras (2009)

Discussiones Mathematicae Graph Theory

A graph is 1-planar if it can be embedded in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree 5 and girth 4 contains (1) a 5-vertex adjacent to an ≤ 6-vertex, (2) a 4-cycle whose every vertex has degree at most 9, (3) a K 1 , 4 with all vertices having degree at most 11.

On locating and differentiating-total domination in trees

Mustapha Chellali (2008)

Discussiones Mathematicae Graph Theory

A total dominating set of a graph G = (V,E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V-S, N(u)∩S ≠ N(v)∩S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V, N[u]∩S ≠ N[v] ∩S. Let γ L ( G ) and γ D ( G ) be the minimum cardinality of a locating-total dominating set and a differentiating-total...

On locating-domination in graphs

Mustapha Chellali, Malika Mimouni, Peter J. Slater (2010)

Discussiones Mathematicae Graph Theory

A set D of vertices in a graph G = (V,E) is a locating-dominating set (LDS) if for every two vertices u,v of V-D the sets N(u)∩ D and N(v)∩ D are non-empty and different. The locating-domination number γ L ( G ) is the minimum cardinality of a LDS of G, and the upper locating-domination number, Γ L ( G ) is the maximum cardinality of a minimal LDS of G. We present different bounds on Γ L ( G ) and γ L ( G ) .

On Longest Cycles in Essentially 4-Connected Planar Graphs

Igor Fabrici, Jochen Harant, Stanislav Jendroľ (2016)

Discussiones Mathematicae Graph Theory

A planar 3-connected graph G is essentially 4-connected if, for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. Jackson and Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that [...] . For a cubic essentially 4-connected planar graph G, Grünbaum with Malkevitch, and Zhang showed that G has a cycle on at least ¾ n vertices. In the present paper the result of Jackson and Wormald is improved. Moreover,...

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