On Hypercube Labellings and Antipodal Monochromatic Paths (original) (raw)

Distance magic labelings of hypercubes

Electronic Notes in Discrete Mathematics, 2013

A distance magic labeling of a graph G is a bijective assignment of labels from {1, 2,. .. , |V (G)|} to the vertices of G such that the sum of labels on neighbors of u is the same for all vertices u. We show that the n-dimensional hypercube has a distance magic labeling for every n ≡ 2 (mod 4). It is known that this condition is also necessary. This completes solution of a conjecture posed by Acharya et al. [2].

The distinguishing number of the hypercube

Discrete Mathematics, 2004

The distinguishing number of a graph G is the minimum number of colors for which there exists an assignment of colors to the vertices of G so that the group of color-preserving automorphisms of G consists only of the identity. It is shown, for the d-dimensional hypercubic graphs H d , that D(H d ) = 3 if d ∈ {2; 3} and D(H d ) = 2 if d ¿ 4. It is also shown that D(H 2 d ) = 4 for d ∈ {2; 3} and D(H 2 d ) = 2 for d ¿ 4, where H 2 d denotes the square of the d-dimensional hypercube. This solves the distinguishing number for hypercubic graphs and their squares.

A note on algebraic hypercube colorings

2008

Abstract An L (1, 1)-coloring of the n-dimensional hypercube Q n assigns nodes of Q n which are at distance les 2 with different colors. Such colorings find application, eg, in frequency assignment in wireless networks and data distribution in parallel memory systems. Let chi 2 macr (Q n) be the minimum number of colors used in any L (1, 1)-coloring. Finding the exact value of chi 2 macr (Q n) is still an open problem, and only 2-approximate solutions are currently known.

Another characterization of hypercubes

Discrete Mathematics, 1982

Nous montrons que dans la classe des graphes connexes tels ;lue deux a&es incidentes quelconques appartiennent a un et un seul quadrilatere, les hypercrtbes finis sont les graphes de degre minimum n lini et possedant 2" sommets.

$D$-Magic and Antimagic Labelings of Hypercubes

arXiv: Combinatorics, 2019

For a set of distances DDD, a graph GGG of order nnn is said to be D−D-D−magic if there exists a bijection f:Vrightarrow1,2,ldots,nf:V\rightarrow \{1,2, \ldots, n\}f:Vrightarrow1,2,ldots,n and a constant kkk such that for any vertex xxx, sumyinND(x)f(y)=k\sum_{y\in N_D(x)} f(y) =ksumyinND(x)f(y)=k, where ND(x)=y∣d(y,x)=j,jinDN_D(x)=\{y|d(y,x)=j, j\in D\}ND(x)=y∣d(y,x)=j,jinD. In this paper we shall find sets of distances DDDs, such that the hypercube is D−D-D−magic. We shall utilise well-known properties of (bipartite) distance-regular graphs to construct the D−D-D−magic labelings.

Routing Permutations in the Hypercube

Lecture Notes in Computer Science, 1999

We study an n-dimensional directed symmetric hypercube Hn, in which every pair of adjacent vertices is connected by two arcs of opposite directions. Using the computer, we show that for H4 and for any permutation on its vertices, there exists a system of pairwise arc-disjoint directed paths from each vertex to its target in the permutation. This gives the answer to Szymanski's conjecture Szy89] for dimension 4. In addition to this study, we consider in Hn the so-called 2-1 routing requests, that is routing requests where any vertex of Hn can be used twice as a source, but only once as a target. We give two such routing requests which cannot be routed in H3. Moreover, we show that for any dimension n 3, it is possible to nd a 2-1 routing request gn such that gn cannot be routed in Hn : in other words, for any n 3, Hn is not (2-1)-rearrangeable.

Hamiltonian cycles and paths in hypercubes with disjoint faulty edges

Inf. Process. Lett., 2021

We consider hypercubes with pairwise disjoint faulty edges. An nnn-dimensional hypercube QnQ_nQn is an undirected graph with 2n2^n2n nodes, each labeled with a distinct binary strings of length nnn. The parity of the vertex is 0 if the number of ones in its labels is even, and is 1 if the number of ones is odd. Two vertices aaa and bbb are connected by the edge iff aaa and bbb differ in one position. If aaa and bbb differ in position iii, then we say that the edge (a,b)(a,b)(a,b) goes in direction iii and we define the parity of the edge as the parity of the end with 0 on the position iii. It was already known that QnQ_nQn is not Hamiltonian if all edges going in one direction and of the same parity are faulty. In this paper we show that if nge4n\ge4nge4 then all other hypercubes are Hamiltonian. In other words, every cube QnQ_nQn, with nge4n\ge4nge4 and disjoint faulty edges is Hamiltonian if and only if for each direction there are two healthy crossing edges of different parity.

On the Existence of Disjoint Spanning Paths in Faulty Hypercubes

Journal of Interconnection Networks, 2010

Assume that n is a positive integer with n ≥ 4 and F is a subset of the edges of the hypercube Qn with |F| ≤ n-4. Let u , x be two distinct white vertices of Qn and v , y be two distinct black vertices of Qn, where black and white refer to the two parts of the bipartition of Qn. Let l1 and l2 be odd integers, where l1 ≥ dQn-F( u , v ), l2 ≥ dQn-F( x , y ), and l1 + l2 = 2n - 2. Moreover, let l3 and l4 be even integers, where l3 ≥ dQn-F( u , x ), l4 ≥ dQn-F( v , y ), and l3+l4 = 2n - 2. In this paper, we prove that there are two disjoint paths P1 and P2 such that (1) P1 is a path joining u to v with length l(P1) = l1, (2) P2 is a path joining x to y with l(P2) = l2, and (3) P1 ∪ P2 spans Qn - F. Moreover, there are two disjoint paths P3 and P4 such that (1) P3 is a path joining u to x with l(P3) = l3, (2) P4 is a path joining v to y with l(P4) = l4, and (3) P3 ∪ P4 spans Qn - F except the following cases: (a) l3 = 2 with dQn-F( u , x ) = 2 and dQn-F-{ v , y }( u , x ) > 2, and (b)...

A survey of the theory of hypercube graphs

Computers & Mathematics With Applications, 1988

Almtract--We present a comprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the n-cube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced paths and cycles. Finally, various embedding and packing problems are discussed, including the determination of the cubical dimension of a given cubical graph.

The Mutually Independent Bipanconnected Property for Hypercube

A graph is denoted by G with the vertex set V(G) and the edge set E(G). A path P = hv 0 , v 1 , . . ., v m i is a sequence of adjacent vertices. Two paths with equal length P 1 = h u 1 , u 2 , . . ., u m i and P 2 = h v 1 , v 2 , . . . , v m i from a to b are independent if u 1 = v 1 = a, u m = v m = b, and u i -v i for 2 6 i 6 m À 1. Paths with equal length fP i g n i¼1 from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let l be a positive integer length, d G (u, v) 6 l 6 jV(G) À 1j with (l À d G (u, v)) being even. We say that the pair of vertices u, v is (m, l)-mutually independent bipanconnected if there exist m mutually independent paths P l i n o m i¼1 with length l from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with d Q n ðu; vÞ P n À 1, is (n À 1, l)mutually independent bipanconnected for every l ; d Q n ðu; vÞ 6 l 6 jVðQ n Þ À 1j with ðl À d Qn ðu; vÞÞ being even. As for d Q n ðu; vÞ 6 n À 2, it is also (n À 1, l)-mutually independent bipanconnected if l P d Q n ðu; vÞ þ 2, and is only (l, l)-mutually independent bipanconnected if l ¼ d Qn ðu; vÞ.