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1995, European Journal of Combinatorics
Graphs that can be isometrically embedded into the metric space 1~ are called /rgraphs. Halved cubes play an important role in the characterization of /i-graphs. We present an algorithm that recognizes halved cubes in O(n log 2 n) time. {~
European Journal of Combinatorics
We fill in the details of the algorithm sketched in [6] and determine its complexity . As a part of this main algorithm , we also describe an algorithm which recognizes graphs which are isometric subgraphs of halved cubes . We discuss possible further applications of the same ideas and give a nice example of non-l 1 -graph allowing a highly isometric embedding into a halved cube .
European Journal of Combinatorics, 1996
We fill in the details of the algorithm sketched in [6] and determine its complexity . As a part of this main algorithm , we also describe an algorithm which recognizes graphs which are isometric subgraphs of halved cubes . We discuss possible further applications of the same ideas and give a nice example of non-l 1 -graph allowing a highly isometric embedding into a halved cube .
Discussiones Mathematicae Graph Theory, 2006
This paper concerns when the complete graph on n vertices can be decomposed into d-dimensional cubes, where d is odd and n is even. (All other cases have been settled.) Necessary conditions are that n be congruent to 1 modulo d and 0 modulo 2 d . These are known to be sufficient for d equal to 3 or 5. For larger values of d, the necessary conditions are asymptotically sufficient by Wilson's results. We prove that for each odd d there is an infinite arithmetic progression of even integers n for which a decomposition exists. This lends further weight to a long-standing conjecture of Kotzig.
Journal of the Australian Mathematical Society, 1996
A graph H decomposes into a graph G if one can write H as an edge-disjoint union of graphs isomorphic to G. H decomposes into D, where D is a family of graphs, when H can be written as a union of graphs each isomorphic to some member of D, and every member of D is represented at least once. In this paper it is shown that the d-dimensional cube Qt decomposes into any graph G of size d each of whose blocks is either an even cycle or an edge. Furthermore, Qj decomposes into D, where D is any set of six trees of size d.
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...
Electronic Notes in Discrete Mathematics, 2006
ABSTRACT Isometric subgraphs of hypercubes are known as partial cubes. These graphs have first been investigated by Graham and Pollak [R.L Graham, H.Pollak On the addressing problem for loop switching, Bell System Technol. J. 50 (1971) 2495–2519] and Djokovic̀ [D. Djokovic̀, Distance preserving subgraphs of the hypercubes, J. Combin. Theory, Ser B41 (1973), 263–267]. Several papers followed with various characterizations of partial cubes. In this paper, we determine all subdivisions of a given configuration which can be embedded isometrically in the hypercube. More specially, we deal with the case where this configuration is a connected graph of order 4 on one hand and the case where the configuration is a fan Fk(k⩾3) on the other hand. Finally, we conjecture that a subdivision of a complete graph of order n(n⩾5) is a partial cube if and only if this one is isomorphic to S(Kn) or there exists n−1 edges of Kn adjacent to a common vertex in the subdivision and the other edges of Kn contain odd added vertices. This proposition is true when the order n∈{4,5,6}.
Discrete Applied Mathematics, 1993
Given two disjoint groups U and V each containing 2 n−1 people. Suppose an experiment (or a game) requires pairwise comparisons between u ∈ U and v ∈ V in such a way that u ∈ U (v ∈ V ) is compared exactly with n people v ∈ V (u ∈ U ). Only one comparison can be done at a time, and the comparisons are made along the n2 n−1 edges of the n−cube graph Q n with vertex classes U and V . The paper deals with sequential orderings of these n2 n−1 pairs (u, v) ∈ U ×V for which the maximum time duration a person has to stay in the sequence for comparison purposes or the total time duration of the 2 n people in the sequence, respectively, is small.
Discrete Mathematics, 2009
For integers n ≥ 1, k ≥ 0, and k ≤ n, the graph Γ k n has vertices the 2 n vectors of F n 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular Γ 1 n is the n-cube, usually denoted by Q n. We examine the binary codes obtained from the adjacency matrices of these graphs when k = 1, 2, 3, following results obtained for the binary codes of the n-cube in Fish [6] and Key and Seneviratne [12]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k = 1, 2, 3, and the dimensions of the ternary codes for k = 1, 2. We also obtain 3-PD-sets for the self-dual binary codes from Γ 2 n when n ≡ 0 (mod 4), n ≥ 8.
Discrete Mathematics, 2009
For integers n ≥ 1, k ≥ 0, and k ≤ n, the graph Γ k n has vertices the 2 n vectors of F n 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular Γ 1 n is the n-cube, usually denoted by Q n. We examine the binary codes obtained from the adjacency matrices of these graphs when k = 1, 2, 3, following results obtained for the binary codes of the n-cube in Fish [6] and Key and Seneviratne [12]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k = 1, 2, 3, and the dimensions of the ternary codes for k = 1, 2. We also obtain 3-PD-sets for the self-dual binary codes from Γ 2 n when n ≡ 0 (mod 4), n ≥ 8.
Lecture Notes in Computer Science
The splitting number of a graph G is the smallest integer k greater than or equal to 0 such that a planar graph can be obtained from G by k splitting operations. Such operation replaces v by two nonadjacent vertices v(1) and v(2), and attaches the neighbors of v either to v(1) or to v(2). The n-cube has a distinguished place in Computer Science. Dean and Richter devoted an article to proving that the minimum number of crossings in an optimum drawing of the 4-cube is 8, but no results about splitting number of other nonplanar n-cubes are known. In this note we give a proof that the splitting number of the 4-cube is 4. In addition, we give the lower bound 2(n-2) for the splitting number of the n-cube. It is known that the splitting number of the n-cube is O(2(n)), thus our result implies that the splitting number of the n-cube is Theta(2n).
22nd International Conference on Data Engineering (ICDE'06), 2006
It is well recognized that data cubing often produces huge outputs. Two popular efforts devoted to this problem are (1) iceberg cube, where only significant cells are kept, and (2) closed cube, where a group of cells which preserve roll-up/drill-down semantics are losslessly compressed to one cell. Due to its usability and importance, efficient computation of closed cubes still warrants a thorough study.
European Journal of Combinatorics, 2007
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).
Discuss. Math. Graph Theory, to appear
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.
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 ...
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.
Canadian mathematical bulletin, 1979
The coarseness, c(G), of a graph G is the maximum number of edge disjoint nonplanar subgraphs contained in G For the n-dimensional cube Q n we obtain the inequalities Introduction. A graph is said to be planar if it can be drawn in the plane (or on a sphere) so that no two of its edges intersect. The coarseness, c(G) of a graph G, a concept introduced first by P. Erdôs, is the maximum number of edge disjoint non-planar subgraphs contained in G. The coarseness of the complete graphs K n and the complete bipartite graphs K mn has been evaluated in [l]-[4], where exact values of c(G) are given in nearly all cases. In the present article we obtain upper and lower bounds for the coarseness of the n-dimensional cube. Some definitions. We adopt the terminology and notation of F. Harary [5]. All graphs considered are finite, undirected and without loops or multiple edges. An edge x = uv of a graph G is called subdivided if it is replaced by a vertex w, called a refinement vertex, and by new edges uw and wv. A graph G' is a subdivision of G, if it is obtained from G by a subdivision of an edge of G. A refinement G of G is a graph obtained from G by a finite sequence of subdivisions. Two graphs are said to be homeomorphic if both can be obtained from the same graph by a sequence of subdivisions of edges. The n-cube Q n is defined inductively as a Cartesian product, where Q 1 = K 2 and Q n = K 2 xQ n-1. A graph isomorphic to a subgraph of Q n is called cubical A refinement G of G which is cubical is called a cubical refinement of G. Since a graph G is planar if and only if c(G) = 0, it follows that c(Q n) = 0 for n = l,2, 3. Main results. Upper and lower bounds are established for c(Q n) in Theorem 1 with the aid of the Lemmas. The number of vertices and edges of a graph G will be denoted by v(G) and e(G) respectively. In particular, u(Q n) = 2 n and e(Q n) = n • 2 n_1 , and each vertex of Q n has degree n.
The Electronic Journal of Combinatorics, 2020
We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from...
Journal of Combinatorial Theory, Series A, 1988
European Journal of Combinatorics
A graph is cubelike if it is a Cayley graph for some elementary abelian 2-group Z n 2. The core of a graph is its smallest subgraph to which it admits a homomorphism. More than ten years ago, Nešetřil andŠámal (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 core and its orbital graphs look very much like cubelike graphs, we are unable to show that this is sufficient to characterise cubelike graphs. However, our results are strong enough to eliminate all non-cubelike vertex-transitive graphs on up to 32 vertices as potential cores of cubelike graphs (of any size). Thus, if one exists at all, a cubelike graph with a non-cubelike core has at least 128 vertices and its core has at least 64 vertices.
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