SIAM Digital Library
 
 
 

SIAM J. on Discrete Mathematics

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue Next Issue

2006

Volume 20, Issue 2, pp. 273-543


Multicolored Hamilton Cycles and Perfect Matchings in Pseudorandom Graphs

Daniela Kühn and Deryk Osthus

SIAM J. Discrete Math. 20, pp. 273-286 (14 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
Given 0 < p < 1, we prove that a pseudorandom graph G with edge density p and sufficiently large order has the following property: Consider any red/blue-coloring of the edges of G and let r denote the proportion of edges which have the color red. Then there is a Hamilton cycle C so that the proportion of red edges of C is close to r. The analogue also holds for perfect matchings instead of Hamilton cycles. We also prove a bipartite version which is used elsewhere to give a minimum-degree condition for the existence of a Hamilton cycle in a 3-uniform hypergraph.

On Multicast Rearrangeable 3-stage Clos Networks Without First-Stage Fan-Out

Hong-Bin Chen and Frank K. Hwang

SIAM J. Discrete Math. 20, pp. 287-290 (4 pages) | Cited 1 time

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
For the multicast rearrangeable 3-stage Clos networks where input crossbars do not have fan-out capability, Kirkpatrick, Klawe, and Pippenger gave a sufficient condition and also a necessary condition which differs from the sufficient condition by a factor of 2. In this paper, we first tighten their conditions. Then we propose a new necessary condition based on the affine plane such that the necessary condition matches the sufficient condition for an infinite class of 3-stage Clos networks.

A Dichotomy Theorem on Fixed Points of Several Nonexpansive Mappings

Tomás Feder

SIAM J. Discrete Math. 20, pp. 291-301 (11 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
The problem of finding a fixed point of a nonexpansive mapping on a hypercube is that it has a polynomial time algorithm. In fact, it is known that one can find a 2-satisfiability characterization of the set of all fixed points in polynomial time. This implies that the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a hypercube is that it has a polynomial time algorithm.
We consider the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a more general Cartesian product of graphs. For a single nonexpansive mapping, a known polynomial time algorithm finds a fixed point and a 2-satisfiability-like characterization of all fixed points. We introduce graphs with a farthest point property (also called apiculate graphs in [H. J. Bandelt and V. Chepoi, The Algebra of Metric Betweenness: Subdirect Representations, Retracts, and Axiomatics, manuscript]), and show that finding a common fixed point of several nonexpansive mappings on Cartesian products of such graphs involves using a polynomial time algorithm. We generalize this result to any family of graphs having a majority function.
By contrast, the smallest graph (in the sense of having the fewest vertices, and the fewest edges of those having the fewest vertices) without the farthest point property is K2,3, and finding a vertex that is a fixed point of two given nonexpansive mappings (retractions) on a Cartesian product of graphs isomorphic to K2,3 is NP-complete. More generally, we exhibit an infinite family of graphs without the farthest point property giving NP-completeness. We show that for any family of graphs not having a majority function, the existence of a common fixed point of two nonexpansive mappings on Cartesian products of such graphs is NP-complete. This proves a dichotomy for the problem based on the existence of a majority function; a similar dichotomy is obtained for the special case of nonexpansive mappings that are retractions. Finally we characterize the families of chordal graphs corresponding to both dichotomies.

Real Number Graph Labellings with Distance Conditions

Jerrold R. Griggs and Xiaohua Teresa Jin

SIAM J. Discrete Math. 20, pp. 302-327 (26 pages) | Cited 6 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
The theory of integer $\lambda$-labellings of a graph, introduced by Griggs and Yeh [J. R. Griggs and R. K.-C. Yeh, SIAM J. Discrete Math., 5 (1992), pp. 586-595], seeks to model efficient channel assignments for a network of transmitters. To prevent interference, labels for nearby vertices must be separated by specified amounts $k_i$ depending on the distance $i$, $1\le i\le p$. Here we expand the model to allow real number labels and separations. The main finding ("D-Set Theorem") is that for any graph, possibly infinite, with maximum degree at most $\Delta$, there is a labelling of minimum span in which all of the labels have the form $\sum_{i=1}^p a_i k_i$, where the $a_i$'s are integers $\ge0$. We show that the minimum span is a continuous function of the $k_i$'s, and we conjecture that it is piecewise linear with finitely many pieces. Our stronger conjecture is that the coefficients $a_i$ can be bounded by a constant depending only on $\Delta$ and $p$. We offer results in strong support of the conjectures, and we give formulas for the minimum spans of several graphs with general conditions at distance two.

A Bound on the Precision Required to Estimate a Boolean Perceptron from Its Average Satisfying Assignment

Paul W. Goldberg

SIAM J. Discrete Math. 20, pp. 328-343 (16 pages) | Cited 2 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
A Boolean perceptron is a linear thresholdfunction over the discrete Boolean domain {0,1}n. That is, it maps any binary vector to 0 or 1, depending on whether the vector's components satisfy some linear inequality. In 1961, Chow showed that any Boolean perceptron is determined by the average or "center of gravity" of its "true" vectors (those that are mapped to 1), together with the total number of true vectors. Moreover, these quantities distinguish the function from any other Boolean function, not just from other Boolean perceptrons.
In this paper we go further, by identifying a lower bound on the Euclidean distance between the average satisfying assignment of a Boolean perceptron and the average satisfying assignment of a Boolean function that disagrees with that Boolean perceptron on a fraction $\epsilon$ of the input vectors. The distance between the two means is shown to be at least $(\epsilon/n)^{O(\log(n/\epsilon)\log(1/\epsilon))}$. This is motivated by the statistical question of whether an empirical estimate of this average allows us to recover a good approximation to the perceptron. Our result provides a mildly superpolynomial upper bound on the growth rate of the sample size required to learn Boolean perceptrons in the "restricted focus of attention" setting. In the process we also find some interesting geometrical properties of the vertices of the unit hypercube.

The Minor Crossing Number

Drago Bokal, Gasper Fijavz, and Bojan Mohar

SIAM J. Discrete Math. 20, pp. 344-356 (13 pages) | Cited 2 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
The minor crossing number of a graph G is defined as the minimum crossing number of all graphs that contain G as a minor. Basic properties of this new invariant are presented. We study topological structure of graphs with bounded minor crossing number and obtain a new strong version of a lower bound based on the genus. We also give a generalization of an inequality of Moreno and Salazar crossing numbers of a graph and its minors.

The Bidimensional Theory of Bounded-Genus Graphs

Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos

SIAM J. Discrete Math. 20, pp. 357-371 (15 pages) | Cited 5 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
Bidimensionality provides a tool for developing subexponential fixed-parameter algorithms for combinatorial optimization problems on graph families that exclude a minor. This paper extends the theory of bidimensionality for graphs of bounded genus (which is a minor-excluding family). Specifically we show that, for any problem whose solution value does not increase under contractions and whose solution value is large on a grid graph augmented by a bounded number of handles, the treewidth of any bounded-genus graph is at most a constant factor larger than the square root of the problem's solution value on that graph. Such bidimensional problems include vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, r-dominating set, connected dominating set, planar set cover, and diameter. On the algorithmic side, by showing that an augmented grid is the prototype bounded-genus graph, we generalize and simplify many existing algorithms for such problems in graph classes excluding a minor. On the combinatorial side, our result is a step toward a theory of graph contractions analogous to the seminal theory of graph minors by Robertson and Seymour.

Classification of Bipartite Boolean Constraint Satisfaction through Delta-Matroid Intersection

Tomás Feder and Daniel Ford

SIAM J. Discrete Math. 20, pp. 372-394 (23 pages) | Cited 1 time

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
Matroid intersection has a known polynomial time algorithm using an oracle. We generalize this result to delta-matroids that do not have equality as a restriction and give a polynomial time algorithm for delta-matroid intersection on delta-matroids without equality using an oracle. We note that when equality is present, delta-matroid intersection is as general as delta-matroid parity. We also obtain algorithms using an oracle for delta-matroid parity on delta-matroids without inequality, and for delta-matroid intersection where one delta-matroid does not contain either equality or inequality, and the second delta-matroid is arbitrary. Both these results also generalize matroid intersection. The results imply a dichotomy for bipartite Boolean constraint satisfaction problems using an oracle when one of the two sides does not contain equality, leaving open cases of delta-matroid parity when both sides have equality; the results also imply a full dichotomy for $k$-partite Boolean constraint satisfaction problems for $k\geq 3$. We then discuss polynomial cases of Boolean constraint satisfaction problems with two occurrences per variable through delta-matroid parity that cannot be obtained using the oracle approach.

The Volume of the Giant Component of a Random Graph with Given Expected Degrees

Fan Chung and Linyuan Lu

SIAM J. Discrete Math. 20, pp. 395-411 (17 pages) | Cited 3 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
We consider the random graph model $G(\mathbf{w})$ for a given expected degree sequence ${\mathbf w} =(w_1, w_2, \ldots, w_n)$. If the expected average degree is strictly greater than $1$, then almost surely the giant component in $G$ of $G({\mathbf w})$ has volume (i.e., sum of weights of vertices in the giant component) equal to $\lambda_0 {\rm Vol}(G) + O(\sqrt{n}\log^{3.5} n)$, where $\lambda_0$ is the unique nonzero root of the equation \[ \sum_{i=1}^n w_i e^{-w_i\lambda} = (1-\lambda) \sum_{i=1}^n w_i, \] and where ${\rm Vol}(G)=\sum_i w_i.$

On the Spanning Ratio of Gabriel Graphs and beta-Skeletons

Prosenjit Bose, Luc Devroye, William Evans, and David Kirkpatrick

SIAM J. Discrete Math. 20, pp. 412-427 (16 pages) | Cited 2 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
The spanning ratio of a graph defined on n points in the Euclidean plane is the maximum ratio over all pairs of data points (u,v) of the minimum graph distance between u and v divided by the Euclidean distance between u and v. A connected graph is said to be an S-spanner if the spanning ratio does not exceed S. For example, for any S there exists a point set whose minimum spanning tree isnot an S-spanner. At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 2.42-spanner [J. M. Keil and C. A. Gutwin, Discrete Comput. Geom., 7 (1992), pp. 13-28]. For proximity graphs between these two extremes, such as Gabriel graphs [K. R. Gabriel and R. R. Sokal, Systematic Zoology, 18 (1969), pp. 259-278], relative neighborhood graphs [G. T. Toussaint, Pattern Recognition, 12 (1980), pp. 261-268], and $\beta$-skeletons [D. G. Kirkpatrick and J. D. Radke, Comput. Geom., G. T. Toussaint, ed., Elsevier, Amsterdam, 1985, pp. 217-248] with $\beta$ in [0,2] some interesting questions arise. We show that the spanning ratio for Gabriel graphs (which are $\beta$-skeletons with $\beta$ = 1) is $\Theta ( \sqrt{n})$ in the worst case. For all $\beta$-skeletons with $\beta$ in [0,1], we prove that the spanning ratio is at most $O(n^\gamma)$, where $\gamma = (1-\log_2(1+\sqrt{1-\beta^2}))/2$. For all $\beta$-skeletons with $\beta$ in [1,2], we prove that there exist point sets whose spanning ratio is at least $\left( \frac{1}{2} - o(1) \right) \sqrt{n} $. For relative neighborhood graphs [G. T. Toussaint, Pattern Recognition, 12 (1980), pp. 261-268] (skeletons with $\beta$ = 2), we show that there exist point sets where the spanning ratio is $\Omega(n)$. For points drawn independently from the uniform distribution on the unit square, we show that the spanning ratio of the (random) Gabriel graph and all $\beta$-skeletons with $\beta$ in [1,2] tends to $\infty$ in probability as $\sqrt{\log n / \log \log n}$.

Full Color Theorems for L(2,1)-Colorings

Peter C. Fishburn and Fred S. Roberts

SIAM J. Discrete Math. 20, pp. 428-443 (16 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
The span $\lambda$ (G) of a graph G is the smallest k for which G's vertices can be L(2,1)-colored, i.e., colored with integers in $\{0,1, \ldots, k \}$ so that adjacent vertices' colors differ by at least 2, and colors of vertices at distance two differ. G is full-colorable if some such coloring uses all colors in $\{0,1, \ldots, \lambda (G) \}$ and no others. We prove that all trees except stars are full-colorable. The connected graph G with the smallest number of vertices exceeding $\lambda$ (G) which is not full-colorable is C6. We describe an array of other connected graphs that are not full-colorable and go into detail on full-colorability of graphs of maximum degree four or less.

A Linear-Time Algorithm for Finding a Maximal Planar Subgraph

Hristo N. Djidjev

SIAM J. Discrete Math. 20, pp. 444-462 (19 pages) | Cited 1 time

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
We construct an optimal linear-time algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G' of G such that adding to G' an extra edge of G results in a nonplanar graph. Our solution is based on a fast data structure for incremental planarity testing of triconnected graphs and a dynamic graph search procedure. Our algorithm can be transformed into a new optimal planarity testing algorithm.

Sparse Sourcewise and Pairwise Distance Preservers

Don Coppersmith and Michael Elkin

SIAM J. Discrete Math. 20, pp. 463-501 (39 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
We introduce and study the notions of pairwise and sourcewise preservers. Given an undirected N-vertex graph G = (V,E) and a set P of pairs of vertices, let G' = (V,H), H \subseteq E, be called a pairwise preserver of G with respect to P if for every pair {u,w} \in P, distG'(u,w) = distG(u,w). For a set S \subseteq V of sources, a pairwise preserver of G with respect to the set of all pairs P = (S \atop 2) of sources is called a sourcewise preserver of G with respect to S. We prove that for every undirected possibly weighted N-vertex graph G and every set P of P = O(N1/2) pairs of vertices of G, there exists a linear-size pairwise preserver of G with respect to P. Consequently, for every subset S \subseteq V of S = O(N1/4) sources, there exists a linear-size sourcewise preserver of G with respect to S. On the negative side we show that neither of the two exponents (1/2 and 1/4) can be improved even when the attention is restricted to unweighted graphs. Our lower bounds involve constructions of dense convexly independent sets of vectors with small Euclidean norms. We believe that the link between the areas of discrete geometry and spanners that we establish is of independent interest and might be useful in the study of other problems in the area of low-distortion embeddings.

On the Greedy Superstring Conjecture

Maik Weinard and Georg Schnitger

SIAM J. Discrete Math. 20, pp. 502-522 (21 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
We investigate the greedy algorithm for the shortest common superstring problem. We show that the length of the greedy superstring is upper-bounded by the sum of the lengths of an optimal superstring and an optimal cycle cover, provided the greedy algorithm happens to merge the strings in a particular way. Thus, when restricting inputs correspondingly, we verify the well-known greedy conjecture, namely, that the approximation ratio of the greedy algorithm is within a factor of two of the optimum, and actually extend the conjecture considerably. We achieve this bound by systematically combining known conditional inequalities about overlaps and period- and string-lengths with a new familiy of string inequalities. We show that conventional systems of conditional inequalities, including the Monge inequalities, are insufficient to obtain our result.

A Note on Unsatisfiable k-CNF Formulas with Few Occurrences per Variable

Shlomo Hoory and Stefan Szeider

SIAM J. Discrete Math. 20, pp. 523-528 (6 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
The (k,s)-SAT problem is the satisfiability problem restricted to instances where each clause has exactly k literals and every variable occurs at most s times. It is known that there exists a function f such that for s \leq f(k) all (k,s)-SAT instances are satisfiable, but (k,f(k)+1)-SAT is already NP-complete (k \geq 3). We prove that f(k) = O(2k \cdot log k/k), improving upon the best known upper bound O(2k/kalpha), where alpha=log3 4 - 1 \approx 0.26. The new upper bound is tight up to a log k factor with the best known lower bound Omega(2k/k).

On Graph Associations

Landon Rabern

SIAM J. Discrete Math. 20, pp. 529-535 (7 pages) | Cited 1 time

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
We introduce a notion of vertex association and consider sequences of these associations. This allows for slick proofs of a few known theorems as well as showing that for any induced subgraph H of G, chi(G) \leq chi(H) + 1/2 (omega (G) + |G| - |H| - 1). As a special case of this, we have chi(G) \leq \lceil omega(G) + tau(G) / 2 \rceil (here chi(G) denotes the chromatic number, omega(G) the clique number, and tau(G) the vertex cover number), which is a generalization of the Nordhaus--Gaddum upper bound. In addition, this settles a conjecture of Reed that chi(G) \leq \lceil omega(G) + Delta(G) + 1 / 2 \rceil in the case when delta(\overline G) \leq omega(\overline G).

Construction of Large Graphs with No Optimal Surjective L(2,1)-Labelings

Daniel Král', Riste Skrekovski, and Martin Tancer

SIAM J. Discrete Math. 20, pp. 536-543 (8 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
An L(2,1)-labeling of a graph G is a mapping c : V(G) \to {0,...,K} such that the labels of two adjacent vertices differ by at least two and the labels of vertices at distance two differ by at least one. A hole of c is an integer h \in {0,...,K} that is not used as a label for any vertex of G. The smallest integer K for which an L(2,1)-labeling of G exists is denoted by lambda(G). The minimum number of holes in an optimal labeling, i.e., a labeling with K = lambda(G), is denoted by rho(G). Georges and Mauro [SIAM J. Discrete Math., 19 (2005), pp. 208-223] showed that rho(G) \le Delta, where Delta is the maximum degree of G, and conjectured that if rho(G) = Delta and G is connected, then the order of G is at most Delta(Delta + 1). We disprove this conjecture by constructing graphs G with rho(G) = Delta and order \lfloor (Delta + 1)2/4 \rfloor (Delta + 1) \approx Delta3/4.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Mean Field Games: Numerical Methods for the Planning Problem
  2. A Convex Condition for Robust Stability Analysis via Polyhedral Lyapunov Functions
  3. Stability and Resolution Analysis for a Topological Derivative Based Imaging Functional
  4. Experimental Design for Biological Systems
  5. Linear Programming and Constrained Average Optimality for General Continuous-Time Markov Decision Pr...
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close