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SIAM J. on Discrete Mathematics

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Issue 4 | 2011 | pp. 1477-1919
Issue 3 | 2011 | pp. 1035-1476
Issue 2 | 2011 | pp. 463-1034 †
Issue 1 | 2011 | pp. 1-461

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2011

Volume 25, Issue 4, pp. 1477-1919


A Note on Bipartite Graph Tiling

Andrzej Czygrinow and Louis DeBiasio

SIAM J. Discrete Math. 25, pp. 1477-1489 (13 pages)

Online Publication Date: November 01, 2011

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Bipartite graph tiling was studied by Zhao [SIAM J. Discrete Math., 23 (2009), pp. 888–900], who gave the best possible minimum degree conditions for a balanced bipartite graph on $2ms$ vertices to contain $m$ vertex disjoint copies of $K_{s,s}$. Let $s<t$ be fixed positive integers. Hladký and Schacht [SIAM J. Discrete Math., 24 (2010), pp. 357–362] gave minimum degree conditions for a balanced bipartite graph on $2m(s+t)$ vertices to contain $m$ vertex disjoint copies of $K_{s,t}$. Their results were best possible, except in the case when $m$ is odd and $t> 2s+1$. We give the best possible minimum degree condition in this case.

Edge-Partitioning Regular Graphs for Ring Traffic Grooming with a Priori Placement of the ADMs

Xavier Muñoz, Zhentao Li, and Ignasi Sau

SIAM J. Discrete Math. 25, pp. 1490-1505 (16 pages)

Online Publication Date: November 01, 2011

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We study the following graph partitioning problem: Given two positive integers $C$ and $\Delta$, find the least integer $M(C,\Delta)$ such that the edges of any graph with maximum degree at most $\Delta$ can be partitioned into subgraphs with at most $C$ edges and each vertex appears in at most $M(C,\Delta)$ subgraphs. This problem is naturally motivated by traffic grooming, which is a major issue in optical networks. Namely, we introduce a new pseudodynamic model of traffic grooming in unidirectional rings, in which the aim is to design a network able to support any request graph with a given bounded degree. We show that optimizing the equipment cost under this model is essentially equivalent to determining the parameter $M(C,\Delta)$. We establish the value of $M(C,\Delta)$ for almost all values of $C$ and $\Delta$, leaving open only the case where $\Delta \geq 5$ is odd, $\Delta \pmod{2C}$ is between $3$ and $C-1$, $C\geq 4$, and the request graph does not contain a perfect matching. For these open cases, we provide upper bounds that differ from the optimal value by at most one.

Computing Geodesic Distances in Tree Space

Megan Owen

SIAM J. Discrete Math. 25, pp. 1506-1529 (24 pages)

Online Publication Date: November 08, 2011

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We present two algorithms for computing the geodesic distance between phylogenetic trees in tree space, as introduced by Billera, Holmes, and Vogtmann [Adv. Appl. Math., 27 (2001), pp. 733–767]. We show that the possible combinatorial types of shortest paths between two trees can be compactly represented by a partially ordered set. We calculate the shortest distance along each candidate path by converting the problem into one of finding the shortest path through a certain region of Euclidean space. In particular, we show there is a linear time algorithm for finding the shortest path between a point in the all-positive orthant and a point in the all-negative orthant of $\mathbb{R}^k$ contained in the subspace of $\mathbb{R}^k$ consisting of all orthants with the first $i$ coordinates nonpositive and the remaining coordinates nonnegative for $0 \leq i \leq k$.

Labeled Ballot Paths and the Springer Numbers

William Y. C. Chen, Neil J. Y. Fan, and Jeffrey Y. T. Jia

SIAM J. Discrete Math. 25, pp. 1530-1546 (17 pages)

Online Publication Date: November 10, 2011

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The Springer numbers are defined in connection with the irreducible root system of type $B_n$ and also arise as the generalized Euler and class numbers introduced by Shanks. Combinatorial interpretations of the Springer numbers have been found by Purtill in terms of André signed permutations, and by Arnol'd in terms of snakes of type $B_n$. We introduce the inversion code of a snake of type $B_n$ and establish a bijection between labeled ballot paths of length $n$ and snakes of type $B_n$. Moreover, we obtain the bivariate generating function for the number $B(n,k)$ of labeled ballot paths starting at $(0,0)$ and ending at $(n,k)$. Using our bijection, we find a statistic $\alpha$ such that the number of snakes $\pi$ of type $B_n$ with $\alpha(\pi)=k$ equals $B(n,k)$. We also show that our bijection specializes to a bijection between labeled Dyck paths of length $2n$ and alternating permutations on $[2n]$.

A Complete Generalization of Clatworthy Group Divisible Designs

Fei Gao and Gennian Ge

SIAM J. Discrete Math. 25, pp. 1547-1561 (15 pages)

Online Publication Date: November 10, 2011

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Partially balanced incomplete block designs (PBIBDs) have a long history and have been extensively used in agriculture and industrial experiments. Since the book of Clatworthy on two-associate-class partially balanced designs was published in 1973, little progress has been made on the construction of these designs. Group divisible designs (GDDs) are an important type of PBIBD with two associate classes. The existence of a GDD with block size $k=3$ was completely settled by Fu, Rodger, and Sarvate. In their works, the most difficult case to solve was when the number of groups, $m$, is less than the block size $k$. The existence of GDDs when $m<k$ is, in general, a difficult case to solve. Indeed, when $k=4$, very little is known about the existence of such GDDs. In this paper, we present two general construction methods for GDDs. The first one is a generalization of Wilson's fundamental construction in combinatorial design theory. The second is an extension of the traditional construction using double group divisible designs. As an application of our new construction methods, a complete solution is provided for GDDs that generalize all eleven designs in the old table of Clatworthy which have block size four, three groups, and replication number at most $10$. For these GDDs, no progress has been made until very recently.

Sublinear Time Algorithms

Ronitt Rubinfeld and Asaf Shapira

SIAM J. Discrete Math. 25, pp. 1562-1588 (27 pages)

Online Publication Date: November 22, 2011

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Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a very small portion of the input. We discuss the types of answers that one can hope to achieve in this setting.

Uniqueness in Discrete Tomography: Three Remarks and a Corollary

Peter Gritzmann, Barbara Langfeld, and Markus Wiegelmann

SIAM J. Discrete Math. 25, pp. 1589-1599 (11 pages)

Online Publication Date: November 22, 2011

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Discrete tomography is concerned with the retrieval of finite point sets in some $\mathbbm{R}^d$ from their X-rays in a given number $m$ of directions $u_1,\ldots, u_m$. In the present paper we focus on uniqueness issues. The first remark gives a uniform treatment and extension of known uniqueness results. In particular, we introduce the concept of $J$-additivity and give conditions when a subset $J$ of possible positions is already determined by the given data. As a by-product, we settle a conjecture of Brunetti and Daurat on planar lattice convex sets. Remark 2 resolves a problem of Kuba posed in 1997 on the uniqueness in the case $d=m=3$ with $u_1,u_2,u_3$ being the standard unit vectors. Remark 3 determines the computational complexity of finding a smallest set $J$ of positions whose disclosure yields uniqueness. As a corollary, we obtain a hardness result for $0$-$1$-polytopes.

Finding Cycles with Topological Properties in Embedded Graphs

Sergio Cabello, Éric Colin de Verdière, and Francis Lazarus

SIAM J. Discrete Math. 25, pp. 1600-1614 (15 pages)

Online Publication Date: November 22, 2011

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Let $G$ be a graph cellularly embedded on a surface $\mathcal{S}$. We consider the problem of determining whether $G$ contains a cycle (i.e., a closed walk without repeated vertices) of a certain topological type in $\mathcal{S}$. We show that the problem can be answered in linear time when the topological type is one of the following: contractible, noncontractible, or nonseparating. In each case, we obtain the same time complexity if we require the cycle to contain a given vertex. On the other hand, we prove that the problem is NP-complete when considering separating or splitting cycles. We also show that deciding the existence of a separating or a splitting cycle of length at most $k$ is fixed-parameter tractable with respect to $k$ plus the genus of the surface.

Asymptotic Study of Subcritical Graph Classes

Michael Drmota, Éric Fusy, Mihyun Kang, Veronika Kraus, and Juanjo Rué

SIAM J. Discrete Math. 25, pp. 1615-1651 (37 pages)

Online Publication Date: December 01, 2011

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We present a unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works in both the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp., $g_n$) of labelled (resp., unlabelled) graphs on $n$ vertices from a subcritical graph class ${\mathcal{G}}=\cup_n {\mathcal{G}_n}$ satisfies asymptotically the universal behavior $g_n = c \!n^{-5/2} \!\gamma^n \! (1+o(1))$ for computable constants $c,\gamma$, e.g., $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that the number of vertices of degree $k$ ($k$ fixed) in a graph chosen uniformly at random from $\mathcal{G}_n$ converges (after rescaling) to a normal law as $n\to\infty$.

A 3/2-Approximation Algorithm for Finding Spanning Trees with Many Leaves in Cubic Graphs

Paul Bonsma and Florian Zickfeld

SIAM J. Discrete Math. 25, pp. 1652-1666 (15 pages)

Online Publication Date: December 01, 2011

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We consider the problem of finding a spanning tree that maximizes the number of leaves (MaxLeaf). We provide a 3/2-approximation algorithm for this problem when restricted to cubic graphs, improving on the previous 5/3-approximation for this class. To obtain this approximation we define a graph parameter $x(G)$ and construct a tree with at least $(n-x(G)+4)/3$ leaves, and we prove that no tree with more than $(n-x(G)+2)/2$ leaves exists. In contrast to previous approximation algorithms for MaxLeaf, our algorithm works with connected dominating sets instead of by constructing a tree directly. The algorithm also yields a 4/3-approximation for the minimum connected dominating set problem in cubic graphs.

Stronger Bounds on Braess's Paradox and the Maximum Latency of Selfish Routing

Henry Lin, Tim Roughgarden, Éva Tardos, and Asher Walkover

SIAM J. Discrete Math. 25, pp. 1667-1686 (20 pages)

Online Publication Date: December 06, 2011

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We give several new upper and lower bounds on the worst-case severity of Braess's paradox and the price of anarchy of selfish routing with respect to the maximum latency objective. In single-commodity networks with arbitrary continuous and nondecreasing latency functions, we prove that this worst-case price of anarchy is exactly $n-1$, where $n$ is the number of network vertices. For Braess's paradox in such networks, we prove that removing at most $c$ edges from a network decreases the common latency incurred by traffic at equilibrium by at most a factor of $c+1$. In particular, the worst-case severity of Braess's paradox with a single edge removal is maximized in Braess's original four-vertex network. In multicommodity networks, we exhibit an infinite family of two-commodity networks, related to the Fibonacci numbers, in which both the worst-case severity of Braess's paradox and the price of anarchy for the maximum latency objective grow exponentially with the network size. This construction demonstrates that numerous known selfish routing results for single-commodity networks have no analogues in networks with two or more commodities. We also prove an upper bound on both of these quantities that is exponential in the network size and independent of the network latency functions, showing that our construction is close to optimal. Finally, we use our family of two-commodity networks to exhibit a natural network design problem with intrinsically exponential (in)approximability.

Boxicity and Poset Dimension

Abhijin Adiga, Diptendu Bhowmick, and L. Sunil Chandran

SIAM J. Discrete Math. 25, pp. 1687-1698 (12 pages)

Online Publication Date: December 08, 2011

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Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times\cdots\times [a_k,b_k]$. The boxicity of $G$, box$(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes; i.e., each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\mathcal{P}=(S,P)$ be a poset, where $S$ is the ground set and $P$ is a reflexive, antisymmetric and transitive binary relation on $S$. The dimension of $\mathcal{P}$, $\dim(\mathcal{P})$, is the minimum integer $t$ such that $P$ can be expressed as the intersection of $t$ total orders. Let $G_{\mathcal{P}}$ be the underlying comparability graph of $\mathcal{P}$; i.e., $S$ is the vertex set and two vertices are adjacent if and only if they are comparable in $\mathcal{P}$. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset $\mathcal{P}$, box$(G_{\mathcal{P}})/(\chi(G_{\mathcal{P}})-1) \le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$, where $\chi(G_{\mathcal{P}})$ is the chromatic number of $G_{\mathcal{P}}$ and $\chi(G_{\mathcal{P}})\ne1$. It immediately follows that if $\mathcal{P}$ is a height-$2$ poset, then box$(G_{\mathcal{P}})\le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$ since the underlying comparability graph of a height-$2$ poset is a bipartite graph. The second result of the paper relates the boxicity of a graph $G$ with a natural partial order associated with the extended double cover of $G$, denoted as $G_c$: Note that $G_c$ is a bipartite graph with partite sets $A$ and $B$ which are copies of $V(G)$ such that, corresponding to every $u\in V(G)$, there are two vertices $u_A\in A$ and $u_B\in B$ and $\{u_A,v_B\}$ is an edge in $G_c$ if and only if either $u=v$ or $u$ is adjacent to $v$ in $G$. Let $\mathcal{P}_c$ be the natural height-$2$ poset associated with $G_c$ by making $A$ the set of minimal elements and $B$ the set of maximal elements. We show that $\frac{\mbox{box}(G)}{2} \le \dim(\mathcal{P}_c) \le 2\mbox{box}(G)+4$. These results have some immediate and significant consequences. The upper bound $\dim(\mathcal{P})\le 2\mbox{box}(G_\mathcal{P})$ allows us to derive hitherto unknown upper bounds for poset dimension such as $\dim(\mathcal{P})\le 2\mbox{ tree width }(G_{\mathcal{P}})+4$, since boxicity of any graph is known to be at most its tree width $+\; 2$. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree $\Delta$ is $O(\Delta\log^2\Delta)$, which is an improvement over the best-known upper bound of $\Delta^2+2$. (2) There exist graphs with boxicity $\Omega(\Delta\log\Delta)$. This disproves a conjecture that the boxicity of a graph is $O(\Delta)$. (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on $n$ vertices with a factor of $O(n^{0.5-\epsilon})$ for any $\epsilon>0$ unless $NP=ZPP$.

Bounds on s-Distance Sets with Strength t

Hiroshi Nozaki and Sho Suda

SIAM J. Discrete Math. 25, pp. 1699-1713 (15 pages)

Online Publication Date: December 08, 2011

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A finite set $X$ in the Euclidean unit sphere is called an $s$-distance set if the set of distances between any two distinct elements of $X$ has size $s$. We say that $t$ is the strength of $X$ if $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. Delsarte Goethals, and Seidel gave an absolute bound for the cardinality of an $s$-distance set. The results of Neumaier and Cameron, Goethals, and Seidel imply that if $X$ is a spherical $2$-distance set with strength $2$, then the known absolute bound for $2$-distance sets can be improved. This bound is also regarded as that for a strongly regular graph with a certain condition of the Krein parameters. In this paper, we give two generalizations of this bound to spherical $s$-distance sets with strength $t$ (more generally, to $s$-distance sets with strength $t$ in a two-point-homogeneous space) and to $Q$-polynomial association schemes. First, for any $s$ and $s-1 \leq t \leq 2s-2$, we improve the known absolute bound for the size of a spherical $s$-distance set with strength $t$. Second, for any $s$, we give an absolute bound for the size of a $Q$-polynomial association scheme of class $s$ with some conditions of the Krein parameters.

A Note on the Maximum Number of Edges of Nonflowerable Coin Graphs

Geir Agnarsson and Jill Bigley Dunham

SIAM J. Discrete Math. 25, pp. 1714-1721 (8 pages)

Online Publication Date: December 08, 2011

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For $n\in\mathbb{N}$ and $4\leq k\leq n$ we compute the exact value of $E_k(n)$, the maximum number of edges of a simple plane graph on $n$ vertices, where each vertex bounds an $\ell$-gon where $\ell\geq k$. The lower bound of $E_k(n)$ is obtained by explicit construction, while the matching upper bound is obtained by solving an integer program by inspection/picture. We then use this result to conjecture the maximum number of edges of a nonflowerable coin graph on $n$ vertices. A flower is a coin graph representation of the wheel graph. A collection of coins or discs in the Euclidean plane is nonflowerable if no flower can be formed by coins from the collection.

A Linear Time Approximation Scheme for Maximum Quartet Consistency on Sparse Sampled Inputs

Sagi Snir and Raphael Yuster

SIAM J. Discrete Math. 25, pp. 1722-1736 (15 pages)

Online Publication Date: December 08, 2011

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Phylogenetic tree reconstruction is a fundamental biological problem. Quartet amalgamation—combining a set of trees over four taxa into a tree over the full set—stands at the heart of many phylogenetic reconstruction methods. This task has attracted many theoretical as well as practical works. However, even reconstruction from a consistent set of quartet trees, i.e., all quartets agree with some tree, is NP-hard, and the best approximation ratio known is $1/3$. For a dense input of $\Theta(n^4)$ quartets that are not necessarily consistent, the problem has a polynomial time approximation scheme. When the number of taxa grows, considering such dense inputs is impractical and some sampling approach is imperative. It is known that given a randomly sampled consistent set of quartets from an unknown phylogeny, one can find, in polynomial time and with high probability, a tree satisfying a $0.425$ fraction of them, an improvement over the $1/3$ ratio. In this paper we further show that given a randomly sampled consistent set of quartets from an unknown phylogeny, where the size of the sample is at least $\Theta(n^2 \log n)$, there is a randomized approximation scheme that runs in linear time in the number of quartets. The previously known polynomial approximation scheme for that problem required a very dense sample of size $\Theta(n^4)$. We note that samples of size $\Theta(n^2 \log n)$ are sparse in the full quartet set. The result is obtained by a combinatorial technique that may be of independent interest.

On the 2-Resonance of Fullerenes

Tomáš Kaiser, Matěj Stehlík, and Riste Škrekovski

SIAM J. Discrete Math. 25, pp. 1737-1745 (9 pages)

Online Publication Date: December 08, 2011

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We show that every pair of hexagons in a fullerene graph satisfying the isolated pentagon rule (IPR) forms a resonant pattern. This solves a problem raised by Ye, Qi, and Zhang [SIAM J. Discrete Math., 23 (2009), pp. 1023–1044].

Graphs with Two Crossings Are 5-Choosable

Zdeněk Dvořák, Bernard Lidický, and Riste Škrekovski

SIAM J. Discrete Math. 25, pp. 1746-1753 (8 pages)

Online Publication Date: December 08, 2011

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A graph $G$ is $k$-choosable if $G$ can be properly colored whenever every vertex has a list of at least $k$ available colors. Thomassen's theorem states that every planar graph is 5-choosable. We extend the result by showing that every graph with at most two crossings is 5-choosable.

Perfect Matchings in Grid Graphs after Vertex Deletions

R. P. Anstee, J. Blackman, and Hangjun Yang

SIAM J. Discrete Math. 25, pp. 1754-1767 (14 pages)

Online Publication Date: December 13, 2011

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We consider the $d$-dimensional grid graph $G=G_m^d$ on vertices $\{1,2,\ldots ,m\}^d$ (a subset of ${\bf Z}^d$), where two vertices are joined if and only if their coordinates differ in one place and have a difference of just 1. The graph is bipartite, and the $m^d$ vertices have bipartition $W$ and $B$ (sets $W$, $B$ can be determined by the parity of their sum of coordinates). We show that there are constants $a_d,b_d$ so that for every even $m$, if we choose subsets $B'\subseteq B$ and $W'\subseteq W$ in the $d$-dimensional grid graph $G$, which satisfy the three conditions (i) $|B'|=|W'|$, (ii) for any $x,y\in B'$, $d_G(x,y)\ge a_dm^{1/d}+b_d$, and (iii) for any $x,y\in W'$, $d_G(x,y)\ge a_dm^{1/d}+b_d$, then $G$ with the vertices $B'\cup W'$ deleted has a perfect matching. The factor $m^{1/d}$ is best possible.

Face Numbers of Certain Cohen–Macaulay Flag Complexes

Jonathan Browder

SIAM J. Discrete Math. 25, pp. 1768-1777 (10 pages)

Online Publication Date: December 13, 2011

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We show that if a $d$-dimensional Cohen–Macaulay complex is, in a certain sense, sufficiently “close” to being balanced, then there is a $d$-dimensional balanced Cohen–Macaulay complex having the same $f$-vector. This in turn provides some partial evidence for a conjecture of Kalai on the $f$-vectors of Cohen–Macaulay flag complexes.
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On Security of Statistical Databases

R. Ahlswede and H. Aydinian

SIAM J. Discrete Math. 25, pp. 1778-1791 (14 pages)

Online Publication Date: December 15, 2011

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A statistical database (SDB) is a database that is used to return statistical information derived from the records to user queries for statistical data analysis. Sometimes, by correlating enough statistics, confidential data (stored in an SDB) about an individual can be inferred. Examples of confidential information stored in an SDB might be salaries or data concerning the medical history of individuals. An important problem is to provide security to SDBs against the disclosure of confidential information. An SDB is said to be secure if no protected data can be inferred from the available queries. One of the security-control methods suggested in the literature consists of query restriction: the security problem is to limit the use of the SDB, introducing a control mechanism, such that no protected data can be obtained from the available queries. Chin and Ozsoyoglu [IEEE Trans. Software Engrg., 8 (1982), pp. 574–582] introduced a control mechanism, called AUDIT EXPERT, where only SUM queries, that is, only certain sums of individual records, are available for the users. This SUM query model leads to several challenging optimization problems. Assume there are $n$ numeric records $\{z_1,\ldots,z_n\}$ stored in database. A natural problem is to maximize the number of answerable SUM queries, that is, the number of subset sums of $\{z_1,\ldots,z_n\}$ (possibly with some additional constraints), that can be returned, such that none of numbers $z_i$ (or sums of subsets of size not exceeding a specified number) can be inferred from these queries. In this paper we give tight bounds for this number under constraints on size and dimension of query subsets.

On Disjoint Common Bases in Two Matroids

Nicholas J. A. Harvey, Tamás Király, and Lap Chi Lau

SIAM J. Discrete Math. 25, pp. 1792-1803 (12 pages)

Online Publication Date: December 15, 2011

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We prove two results on packing common bases of two matroids. First, we show that the computational problem of common base packing reduces to the special case where one of the matroids is a direct sum of uniform matroids. Second, we give a counterexample to a conjecture of Chow, which proposed a sufficient condition for the existence of a common base packing. Chow's conjecture is a generalization of Rota's basis conjecture.

Covering a Graph by Forests and a Matching

Tomáš Kaiser, Mickaël Montassier, and André Raspaud

SIAM J. Discrete Math. 25, pp. 1804-1811 (8 pages)

Online Publication Date: December 15, 2011

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We prove that for any positive integer $k$, the edges of any graph whose fractional arboricity is at most $k + 1/(3k+2)$ can be decomposed into $k$ forests and a matching. This is a partial result in the direction of the “Nine Dragon Tree” conjecture of Montassier et al.

Proof of the Goresky Klapper Conjecture on Decimations of $L$-sequences

Todd Cochrane and Sergei Konyagin

SIAM J. Discrete Math. 25, pp. 1812-1831 (20 pages)

Online Publication Date: December 15, 2011

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Let $p$ be an odd prime and $\mathbb E=\{2,4, \dots, p-1\}$ the set of nonzero even residues in $\mathbb Z_p = \mathbb Z/(p)$. We prove that for $p>13$, if the mapping $x \to Ax^k$ is a permutation of $\mathbb Z_p$, but not the identity mapping, then the mapping is not a permutation of $\mathbb E$. This establishes a conjecture of Goresky and Klapper stating that any two distinct decimations of a binary $\ell$-sequence are cyclically distinct.

On Universal Cycles for new Classes of Combinatorial Structures

Antonio Blanca and Anant P. Godbole

SIAM J. Discrete Math. 25, pp. 1832-1842 (11 pages)

Online Publication Date: December 20, 2011

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A universal cycle (u-cycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of u-cycles for collections of subsets, restricted multisets, and lattice paths. For subsets, we show that a u-cycle exists for the $k$-subsets of an $n$-set if we let $k$ vary in a non zero length interval. We use this result to construct a “covering” of length $(1+o(1))$$n \choose k$ for all subsets of $[n]$ of size exactly $k$ with a specific formula for the $o(1)$ term. We also show that u-cycles exist for all $n$-length words over some alphabet $\Sigma,$ which contain all characters from $R \subset \Sigma.$ Using this result we provide u-cycles for encodings of Sperner families of size 2 and proper chains of subsets.

Broken Bracelets, Molien Series, Paraffin Wax, and an Elliptic Curve of Conductor 48

Tewodros Amdeberhan, Mahi̇r Bi̇len Can, and Victor H. Moll

SIAM J. Discrete Math. 25, pp. 1843-1859 (17 pages)

Online Publication Date: December 20, 2011

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This paper introduces the concept of necklace binomial coefficients motivated by the enumeration of a special type of sequences. Several properties of these coefficients are described, including a connection between their roots and an elliptic curve. Further links are given to a physical model from quantum mechanical supersymmetry as well as properties of alkane molecules in chemistry.

Deriving Finite Sphere Packings

Natalie Arkus, Vinothan N. Manoharan, and Michael P. Brenner

SIAM J. Discrete Math. 25, pp. 1860-1901 (42 pages)

Online Publication Date: December 20, 2011

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Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of $n$ spheres in $\mathbb{R}^3$ satisfying minimal rigidity constraints ($\geq 3$ contacts per sphere and $\geq 3n-6$ total contacts). We derive such packings for $n \leq 10$ and provide a preliminary set of maximum contact packings for $10 < n \leq 20$. The resultant set of packings has some striking features; among them are the following: (i) all minimally rigid packings for $n \leq 9$ have exactly $3n-6$ contacts; (ii) nonrigid packings satisfying minimal rigidity constraints arise for $n \geq 9$; (iii) the number of ground states (i.e., packings with the maximum number of contacts) oscillates with respect to $n$; (iv) for $10 \leq n \leq 20$ there are only a small number of packings with the maximum number of contacts, and for $10 \leq n < 13$ these are all commensurate with the hexagonal close-packed lattice. The general method presented here may have applications to other related problems in mathematics, such as the Erdös repeated distance problem and Euclidean distance matrix completion problems.

Blocks and Cut Vertices of the Buneman Graph

A. W. M. Dress, K. T. Huber, J. Koolen, and V. Moulton

SIAM J. Discrete Math. 25, pp. 1902-1919 (18 pages)

Online Publication Date: December 20, 2011

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Given a set $\Sigma$ of bipartitions of some finite set $X$ of cardinality at least $2$, one can associate to $\Sigma$ a canonical $X$-labeled graph $\mathcal{B}(\Sigma)$, called the Buneman graph. This graph has several interesting mathematical properties—for example, it is a median network and therefore an isometric subgraph of a hypercube. It is commonly used as a tool in studies of DNA sequences gathered from populations. In this paper, we present some results concerning the cut vertices of $\mathcal{B}(\Sigma)$, i.e., vertices whose removal disconnect the graph, as well as its blocks or $2$-connected components—results that yield, in particular, an intriguing generalization of the well-known fact that $\mathcal{B}(\Sigma)$ is a tree if and only if any two splits in $\Sigma$ are compatible.
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  1. Testing Triangle-Freeness in General Graphs
  2. Combinatorial Properties of a Rooted Graph Polynomial
  3. Locating Servers for Reliability and Affine Embeddings
  4. Approximation Algorithms for Network Design with Metric Costs
  5. Correlation of Graph‐Theoretical Indices
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