Homogeneous Embeddings of Cycles in Graphs (original) (raw)
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Some general aspects of the framing number of a digraph
Discrete Applied Mathematics, 1998
A digraph D is homogeneously embedded in a digraph H if for each vertex x of D and each vertex J' of H. there exists an embedding of D in H as an induced subdigraph with .I at J'. A digraph F of minimum order in which D can be homogeneously embedded is called a frame of D and the order of F is called the framing number of D. Several general results involving frames and framing numbers of digraphs arc established. The framing number is determined for a number of classes of digraphs, including a class of digraphs whose underlying graph is a complete bipartite graph, a class of digraphs whose underlying graph is C, + k'l. and the lexicographic product of a transitive tournament and a vertex transitive digraph. A relationship between the diameters of the underlying graphs of a digraph and its frame is determined. We show that every tournament has a frame which is also a tournament. 0 199X Elsevicr Science B.V. All rights reserved.
Embedding index in some classes of graphs
A Subset S of the vertex set of a graph G is called a dominating set of G if each vertex of G is either in S or adjacent to at least one vertex in S. A partition D = {D 1 , D 2 , …, D k } of the vertex set of G is said to be a domatic partition or simply a d-partition of G if each class D i of D is a dominating set in G. The maximum cardinality taken over all d-partitions of G is called the domatic number of G denoted by d (G). A graph G is said to be domatically critical or d-critical if for every edge x in G, d (G -x) < d (G), otherwise G is said to be domatically non d-critical. The embedding index of a non d-critical graph G is defined to be the smallest order of a d-critical graph H containing G as an induced subgraph denoted by ) (G θ . In this paper, we find the ) (G θ for the Barbell graph, the Lollipop graph and the Tadpole graph.
A Subset S of the vertex set of a graph G is called a dominating set of G, if each vertex of G is either in S or adjacent to at least one vertex in S. A partition D = D 1 D 2 D k of the vertex set of G is said to be a domatic partition or simply a d-partition of G, if each class of D i of D is a dominating set in G. The maximum cardinality taken over all d-partitions of G is called the domatic number of G denoted by d G . A graph G is said to be domatically critical or d-critical if for every edge x in G, d G − x < d G otherwise G is said to be domatically non d-critical. The embedding index of a non d-critical graph G is defined to be the smallest order of d-critical graph H containing G as an induced sub graph denoted by G . In this paper, we find the upper bound of G for few well known classes of graphs.
Proceedings of the Edinburgh Mathematical Society, 1981
We show that every simple graph of order 2r and minimum degree ≧4r/3 has the property that for any partition of its vertex set into 2-subsets, there is a cycle which contains exactly one vertex from each 2-subset. We show that the bound 4r/3 cannot be lowered to r, but conjecture that it can be lowered to r + 1.
Interchangeability of relevant cycles in graphs
The set R of relevant cycles of a graph G is the union of its minimum cycle bases. We introduce a partition of R such that each cycle in a class W can be expressed as a sum of other cycles in W and shorter cycles. It is shown that each minimum cycle basis contains the same number of representatives of a given class W. This result is used to derive upper and lower bounds on the number of distinct minimum cycle bases. Finally, we give a polynomial-time algorithm to compute this partition.
Embedding of cycles in arrangement graphs
IEEE Transactions on Computers, 1993
Arrangement graphs have been recently proposed as an attractive interconnection topology for large multiprocessor systems. In this correspondence, we further study these graphs by first proving the existence of Hamiltonian cycles in any arrangement graph. Secondly, we prove that an arrangement graph contains cycles of all lengths ranging between 3 and the size of the graph. Finally, we show that an arrangement graph can be decomposed into node disjoint cycles in many different ways. Index Tem-4rrangement graphs, disjoint cycles, embeddings, Hamiltonian cycles, interconnection networks, star graphs.
Cycle-saturated graphs of minimum size
Discrete Mathematics, 1996
A graph G is called Ck-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in Ck-saturated graphs for all k ~ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + cln/k and n + c2n/k for some positive constants cl and c2. Our results provide an asymptotic solution to a 15-year-old problem of Bollob~,s.
On the -labelings of amalgamations of graphs
Discrete Applied Mathematics, 2013
The problem of assigning frequencies to transmitters in a radio network can be modeled through vertex labelings of a graph, wherein each vertex represents a transmitter and edges connect vertices whose corresponding transmitters are operating in close proximity. In one such model, an L(2, 1)-labeling of a graph G is employed, which is an assignment f of nonnegative integers to the vertices of G such that if vertices x and y are adjacent,
Consistent Cycles in Graphs and Digraphs
Graphs and Combinatorics, 2007
Let Γ be a finite digraph and let G be a subgroup of the automorphism group of Γ. A directed cycle C of Γ is called G-consistent whenever there is an element of G whose restriction to C is the 1-step rotation of C. Consistent cycles in finite arc-transitive graphs were introduced by Conway in one of his public lectures. He observed that the number of G-orbits of G-consistent cycles of an arc-transitive group G is precisely one less than the valency of the graph. In this paper, we give a detailed proof of this result in a more general settings of arbitrary groups of automorphisms of graphs and digraphs.