Subdivided graphs as isometric subgraphs of Hamming graphs (original) (raw)

On the Complexity of Recognizing Hamming Graphs and Related Classes of Graphs

European Journal of Combinatorics, 1996

This paper contains a new algorithm that recognizes whether a given graph G is a Hamming graph , i . e . a Cartesian product of complete graphs , in O ( m ) time and O ( n 2 ) space . Here m and n denote the numbers of edges and vertices of G , respectively . Previously this was only possible in O ( m log n ) time .

On k-partitioning of Hamming graphs

Discrete Applied Mathematics, 1999

We consider the graphs H n a de ned as the Cartesian products of n complete graphs with a vertices each. Let an edge cut partition the vertex set of a graph into k subsets A 1 ; : : : ; A k with jjA i j ? jA j jj 1. We consider the problem of determining the minimal size of such a cut for the graphs de ned above and present bounds and asymptotic results for some speci c values of k.

Isometric embeddings of subdivided connected graphs into hypercubes

Discrete Mathematics, 2009

Isometric subgraphs of hypercubes are known as partial cubes. These graphs have first been investigated by Graham and Pollack [R.L. Graham, H. Pollack, On the addressing problem for loop switching, Bell System Technol. J. 50 (1971) 2495-2519; and D. Djoković, Distance preserving subgraphs of hypercubes, J. Combin. Theory Ser. B 14 (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 specifically, we deal with the case where this configuration is a connected graph of order 4, a complete graph of order 5 and the case of a k-fan F k (k ≥ 3).

Isometric embedding of subdivided Connected graphs in the hypercube

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}.

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...

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.

The number of edges in a subgraph of a Hamming graph

Applied Mathematics Letters, 2001

G be a subgraph of the Cartesian product Hamming graph (Kp)r with n vertices. Then the number of edges of G is at most (1/2)(p -1) log, n, with equality holding if and only G is isomorphic to (Kp)s for some s 5 r.

The competition numbers of Hamming graphs with diameter at most three

2010

The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs. In this paper, we compute the competition numbers of Hamming graphs with diameter at most three.