On Θ-graphs of partial cubes (original) (raw)
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On Theta-graphs of partial cubes
2007
The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K 1 by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ -graphs.
Three quotient graphs factored through the Djokovic-Winkler relation
For a partial cube (that is, an isometric subgraph of a hypercube) G, quotient graphs G # , G τ , and G ∼ have the equivalence classes of the Djoković-Winkler relation as the vertex set, while edges are defined in three different natural ways. Several results on these quotients are proved and the concepts are compared. For instance, for every graph G there exists a median graph M such that G = M τ . Triangle-free and complete quotient graphs are treated and it is proved that for a median graph G its τ -graph is triangle-free if and only if G contains no convex K 1,3 . Connectedness and the question of when quotients yield the same graphs are also treated.
On the k-subgraphs of the generalized n-cubes
Graphs are used in modeling interconnections networks and measuring their properties. Knowing and understanding the graph theoretical/combinatorial properties of the underlying networks are necessary in developing more efficient parallel algorithms as well as fault-tolerant communication/routing algorithms [1] The hypercube is one of the most versatile and efficient networks yet discovered for parallel computation. One generalization of the hypercube is the n-cube Q(n,m) which is a graph whose vertices are all the binary n-tuples, such that two vertices are adjacent whenever they differ in exactly m coordinates. The k-subgraph of the Generalized n-cube Q k (n,m) is the induced subgraph of the n-cube Q(n,m) where q=2, such that a vertex v ∈ V(Q k (n,m)) if and only if v ∈ V(Q(n,m)) and v is of parity k. This paper presents some degree properties of Q k (n,m) as well as some isomorphisms it has with other graphs, namely: 1)) 2 , (1 n Q n− is isomorphic to Kn 2)) 2 , (i n Q k is isomor...
arXiv: Combinatorics, 2016
We study a family of graphs related to the n-cube. The middle cube graph of parameter k is the subgraph of Q 2k−1 induced by the set of vertices whose binary representation has either k − 1 or k number of ones. The middle cube graphs can be obtained from the wellknown odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors).
Partial cubes as subdivision graphs and as generalized Petersen graphs
Discrete Mathematics, 2003
Isometric subgraphs of hypercubes are known as partial cubes. The subdivision graph of a graph G is obtained from G by subdividing every edge of G. It is proved that for a connected graph G its subdivision graph is a partial cube if and only if every block of G is either a cycle or a complete graph. Regular partial cubes are also considered. In particular it is shown that among the generalized Petersen graphs P (10, 3) and P (2n, 1), n ≥ 2, are the only (regular) partial cubes.
Partial cubes: structures, characterizations, and constructions
Discrete Mathematics, 2008
Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djoković's and Winkler's relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated.
On regular subgraphs of augmented cubes
AKCE International Journal of Graphs and Combinatorics, 2020
The n-dimensional augmented cube AQ n is a variation of the hypercube Q n : It is a ð2n À 1Þ-regular and ð2n À 1Þ-connected graph on 2 n vertices. One of the fundamental properties of AQ n is that it is pancyclic, that is, it contains a cycle of every length from 3 to 2 n : In this paper, we generalize this property to k-regular subgraphs for k ¼ 3 and k ¼ 4: We prove that the augmented cube AQ n with n ! 4 contains a 4-regular, 4-connected and pancyclic subgraph on l vertices if and only if 8 l 2 n : Also, we establish that for every even integer l from 4 to 2 n , there exists a 3-regular, 3-connected and pancyclic subgraph of AQ n on l vertices.
C O ] 6 A ug 2 01 8 Cores of Cubelike Graphs
2018
A graph is cubelike if it is a Cayley graph for some elementary abelian 2-group Zn 2 . The core of a graph is its smallest subgraph to which it admits a homomorphism. More than ten years ago, Nešetřil and Šámal (On tension-continuous mappings. European J. Combin., 29(4):1025–1054, 2008) asked whether the core of a cubelike graph is cubelike, but since then very little progress has been made towards resolving the question. Here we investigate the structure of the core of a cubelike graph, deducing a variety of structural, spectral and group-theoretical properties that the core “inherits” from the host cubelike graph. These properties constrain the structure of the core quite severely — even if the core of a cubelike graph is not actually cubelike, it must bear a very close resemblance to a cubelike graph. Moreover we prove the much stronger result that not only are these properties inherited by the core of a cubelike graph, but also by the orbital graphs of the core. Even though the ...
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