Showing papers in "Ars Combinatoria in 2010"

Pattern avoidance in set partitions

Abstract: The study of patterns in permutations is a very active area of current research. Klazar dened and studied an analogous notion of pattern for set partitions. We continue this work, nding exact formulas for the number of set partitions which avoid certain specic patterns. In particular, we enumerate and characterize those partitions avoiding any partition of a 3-element set. This allows us to conclude that the corresponding sequences are P-recursive. Finally, we dene a second notion of pattern in a set partition, based on its restricted growth function. Related results are obtained for this new denition.

The generalized connectivity of complete bipartite graphs

Abstract: Let G be a nontrivial connected graph of order n, and k an integer with 2 � kn. For a set S of k vertices of G, let �(S) denote the maximum numberof edge-disjoint trees T1;T2;:::;Tin G such that V (Ti) \ V (Tj) = S for every pair i;j of distinct integers with 1 � i;j� `. Chartrand et al. generalized the concept of connectivity as follows: The k-c潮nectivity , denoted byk(G), of G is defined byk(G) =minf�(S)g, where the minimum is taken over all k-subsets S of V (G). Thus �2(G) = �(G), where �(G) is the connectivity of G. Moreover, �n(G) is the maximum number of edge-disjoint spanning trees of G. This paper mainly focus on the k-connectivity of complete bipartite graphs Ka;b. First, we obtain the number of edge-disjoint spanning trees of Ka;b, which is b ab a+b−1 c, and specifically give the b ab a+b−1 c edge-disjoint spanning trees. Then based on this result, we get the k-connectivity of Ka;bfor all 2 � ka+b. Namely, if k > b−a+2

On Super and Restricted Connectivity of Some Interconnection Networks.

Abstract: The super (resp., edge-) connectivity of a connected graph is the minimum cardinality of a vertex-cut (resp., an edge-cut) whose removal does not isolate a vertex. In this paper, we consider the two parameters for a special class of graphs G(G0, G1; M), proposed by Chen et al [Applied Math. and Computation, 140 (2003), 245-254], obtained from two k-regular k-connected graphs G0 and G1 with the same order by adding a perfect matching between their vertices. Our results improve ones of Chen et al. As applications, the super connectivity and the super edge-connectivity of the n-dimensional hypercube, twisted cube, cross cube, Möbius cube and locally twisted cube are all 2n − 2.

Genus Distribution of Graph Amalgamations: Pasting at Root-Vertices.

Abstract: We pursue the problem of counting the imbeddings of a graph in each of the orientable surfaces. We demonstrate how to achieve this for an iterated amalgamation of arbitrarily many copies of any graph whose genus distribution is known and further analyzed into a partitioned genus distribution. We introduce the concept of recombinant strands of face-boundary walks, and we develop the use of multiple production rules for deriving simultaneous recurrences. These two ideas are central to a broadbased approach to calculating genus distributions for graphs synthesized from smaller graphs.

On The Generalized Fibonacci And Pell Sequences By Hessenberg Matrices.

Abstract: In this paper, we consider the generalized Fibonacci and Pell Sequences and then show the relationships between the generalized Fibonacci and Pell sequences, and the Hessenberg permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Pell Sequence, fPng ; is de…ned by the recurrence relation, for n 1 Pn+1 = 2Pn + Pn 1 (1.2) where P0 = 0; P1 = 1: The well-known Fibonacci and Pell numbers can be generalized as follow: Let A be nonzero, relatively prime integers such that D = A +4 6= 0: De…ne sequence fung by, for all n 2 (see [17]), un = Aun 1 + un 2 (1.3) where u0 = 0; u1 = 1: If A = 1; then un = Fn (the nth Fibonacci number). If A = 2; then un = Pn ( the nth Pell number). An alternative is to let the roots of the equation t At 1 = 0 be, for n 0 un = n n : The sequence fung have studied by several authors (see [6], [1]). The following identities can be found in [6], [1]:

A survey of some open questions in reconstruction numbers.

Abstract: Frank Harary contributed numerous questions to a variety of topics in graph theory. One of his favourite topics was the Reconstruction Problem which, in its first issue in 1977, the Journal of Graph Theory described as the major unsolved problem in the field. Together with Plantholt, Frank Harary initiated the study of reconstruction numbers of graphs. We shall here present a survey of some of the work done on reconstruction numbers, focusing mainly on the questions which this work leaves open.

On short zero-sum subsequences over p-groups

Abstract: Let G be a finite abelian group with exponent n. Let s(G) denote the smallest integer l such that every sequence over G of length at least l has a zero-sum subsequence of length n. For p-groups whose exponent is odd and sufficiently large (relative to Davenport’s constant of the group) we obtain an improved upper bound on s(G), which allows to determine s(G) precisely in special cases. Our results contain Kemnitz’ conjecture, which was recently proved, as a special case.

Negativity Subscripted Fibonacci And Lucas Numbers And Their Complex Factorizations.

Abstract: In this paper, we …nd families of (0; 1; 1) tridiagonal matrices whose determinants and permanents equal to the negatively subscripted Fibonacci and Lucas numbers. Also we give complex factorizations of these numbers by the …rst and second kinds of Chebyshev polynomials. 1. Introduction The well-known Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 2 Fn+1 = Fn + Fn 1 (1.1) where F1 = F2 = 1: The Lucas Sequence, fLng ; is de…ned by the recurrence relation, for n 2 Ln+1 = Ln + Ln 1 (1.2) where L1 = 1; L2 = 3: Rules (1.1) and (1.2) can be used to extend the sequence backward, respectively, thus F 1 = F1 F0; F 2 = F0 F 1 L 1 = L1 L0; L 2 = L0 L 1; : : : ; and so on. Clearly F n = F n+2 F n+1 = ( 1) Fn; (1.3) L n = L n+2 L n+1 = ( 1) Ln: (1.4) In [9] and [5], the authors give complex factorizations of the Fibonacci numbers by considering the roots of Fibonacci polynomials as follows

Fibonacci Graceful Graphs.

Abstract: assigns each edge a different label. The problem of characterizing all graceful graphs remains open (Golomb [3]), and in particular the Ringel-Kotzig-Rosa conjecture that all trees are graceful is still unproved after fifteen years. (For a summary of the status of this conjecture, see Bloom [2].) Other classes of graphs that are known to be graceful include complete bipartite graphs (Rosa [7]), wheels (Hoede & Kuiper [5]), and cycles on n vertices where n = 0 or 3 (mod 4) (Hebbare [4]).

On The Generalized Lucas Sequences by Hessenberg Matrices.

Abstract: We show that there are relationships between a generalized Lucas sequence and the permanent and determinant of some Hessenberg matrices.

On a conjecture about inverse domination in graphs

Abstract: Let G = (V,E) be a graph with no isolated vertex. A classical observation in domination theory is that if D is a minimum dominating set of G, then V \D is also a dominating set of G. A set D′ is an inverse dominating set of G if D′ is a dominating set of G and D′ ⊆ V \D for some minimum dominating set D of G. The inverse domination number of G is the minimum cardinality among all inverse dominating sets of G. The independence number of G is the maximum cardinality of an independent set of vertices in G. Domke, Dunbar, and Markus (Ars Combin. 72 (2004), 149–160) conjectured that the inverse domination number of G is at most the independence number of G. We prove this conjecture for special families of graphs, including claw-free graphs, bipartite graphs, split graphs, very well covered graphs, chordal graphs and cactus graphs.

The Laplacian spectral radius of graphs with given matching number

Abstract: Abstract In this paper, we show that of all graphs of order n with matching number β, the graphs with maximal spectral radius are Kn if n = 2β or 2β + 1; K 2 β + 1 ∪ K n - 2 β - 1 ¯ if 2β + 2 ⩽ n K β ⋁ K n - β ¯ or K 2 β + 1 ∪ K n - 2 β - 1 ¯ if n = 3β + 2; K β ⋁ K n - β ¯ if n > 3β + 2, where K t ¯ is the empty graph on t vertices.

Algebraic Properties and Panconnectivity of Folded Hypercubes.

Abstract: This paper considers the folded hypercube FQn, as an enhancement on the hypercube, and obtain some algebraic properties of FQn. Using these properties the authors show that for any two vertices x and y in FQn with distance d and any integers h ∈ {d, n + 1 − d} and l with h ≤ l ≤ 2 − 1, FQn contains an xy-path of length l and no xy-paths of other length provided that l and h have the same parity.

Note on the Rainbow k-Connectivity of Regular Complete Bipartite Graphs

Abstract: A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of the path are colored the same. For a �-connected graph G and an integer k with 1 � k � �, the rainbow kconnectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edge-coloring of G such that any two distinct vertices of G are connected by k internally disjoint rainbow paths. Denote by Kr,r an r-regular complete bipartite graph. Chartrand et al. in “G. Chartrand, G.L. Johns, K.A. McKeon, P. Zhang, The rainbow connectivity of a graph, Networks 54(2009), 75-81” left an open question of determining an integer g(k) for which the rainbow k-connectivity of Kr,r is

Flag-transitive 2-(v, k, 4) symmetric designs.

Abstract: In this article, we study the classification of flag-transitive, point-primitive 2- (v, k, 4) symmetric designs. We prove that if the socle of the automorphism group G of a flag-transitive, point-primitive nontrivial 2- (v, k, 4) symmetric design is an alternating group An for n≥5, then (v, k) = (15, 8) and is one of the following: (i) The points of are those of the projective space PG(3, 2) and the blocks are the complements of the planes of PG(3, 2), G = A7 or A8, and the stabilizer Gx of a point x of is L3(2) or AGL3(2), respectively. (ii) The points of are the edges of the complete graph K6 and the blocks are the complete bipartite subgraphs K2, 4 of K6, G = A6 or S6, and Gx = S4 or S4 × Z2, respectively. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:475-483, 2011

Domination and Total Domination Contraction Numbers of Graphs.

Abstract: In this paper we consider the effect of edge contraction on the domination number and total domination number of a graph. We define the (total) domination contraction number of a graph as the minimum number of edges that must be contracted in order to decrease the (total) domination number. We show both of this two numbers are at most three for any graph. In view of this result, we classify graphs by their (total) domination contraction numbers and characterize these classes of graphs.

On the number of points of a hypersurface in finite projective space (after J.-P. Serre)

Abstract: In J.-P. Serre's Lettre et M. Tsfasman [3], an interesting bound for the maximal number of points on a hypersurface of the n-dimensional projective space PG(n, q) over the Galois field GF(q) with q elements is given. Using essentially the same combinatorial technique as in [3], we provide a bound which is relative to the maximal dimension of a subspace of PG(n, q) which is completely contained in the hypersurface. The lower that dimension, the better the bound. Next, by using a different argument, we derive a bound which is again relative to the maximal dimension of a subspace of PG(n, q) which is completely contained in the hypersurface. If that dimension increases for the latter case, the bound gets better. As such, the bounds are complementary.

On Families Of Bipartite Graphs Associated With Sums Of Generalized Order-k Fibonacci And Lucas Numbers.

Abstract: In this paper, we consider the relationships between the sums of the generalized order-k Fibonacci and Lucas numbers and 1-factors of bipartite graphs. 1. Introduction We consider the generalized order k Fibonacci and Lucas numbers. In [1], Er de…ned k sequences of the generalized order k Fibonacci numbers as shown: g n = k X j=1 g n j ; for n > 0 and 1 i k; (1.1) with boundary conditions for 1 k n 0; g n = 1 if i = 1 n; 0 otherwise, where g n is the nth term of the ith sequence. For example, if k = 2, then g n is usual Fibonacci sequence, fFng ; and, if k = 4, then the 4th sequence of the generalized order 4 Fibonacci numbers is 1; 1; 2; 4; 8; 15; 29; 56; 108; 208; 401; 773; 1490; : : : : In [9], the authors de…ned k sequences of the generalized order k Lucas numbers as shown: l n = k X j=1 l n j , for n > 0 and 1 i k, (1.2) 2000 Mathematics Subject Classi…cation. 11B39, 15A15, 15A36, 05C50.