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2012

Volume 26, Issue 1, pp. 1-414


Forbidden Induced Subgraphs of Double-split Graphs

Boris Alexeev, Alexandra Fradkin, and Ilhee Kim

SIAM J. Discrete Math. 26, pp. 1-14 (14 pages)

Online Publication Date: January 03, 2012

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In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic classes are closed under the operation of taking induced subgraphs (and have known forbidden subgraph characterizations), while the fifth one, consisting of double-split graphs, is not. A graph is doubled if it is an induced subgraph of a double-split graph. We find the forbidden induced subgraph characterization of doubled graphs; it contains 44 graphs.

A Deterministic Algorithm for the Frieze–Kannan Regularity Lemma

Domingos Dellamonica, Subrahmanyam Kalyanasundaram, Daniel Martin, Vojtěch Rödl, and Asaf Shapira

SIAM J. Discrete Math. 26, pp. 15-29 (15 pages)

Online Publication Date: January 03, 2012

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The Frieze–Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. R. Williams recently asked if one can construct a partition satisfying the conditions of the Frieze–Kannan regularity lemma in deterministic subcubic time. We resolve this problem by designing an $\tilde O(n^{\omega})$ time algorithm for constructing such a partition, where $\omega < 2.376$ is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze–Kannan regularity lemma.

Binary Nontiles

Don Coppersmith and Victor S. Miller

SIAM J. Discrete Math. 26, pp. 30-38 (9 pages)

Online Publication Date: January 17, 2012

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A subset $V \subseteq \mathbb{F}_2^n$ is a tile if $\mathbb{F}_2^n$ can be covered by disjoint translates of $V$. In other words, $V$ is a tile if and only if there is a subset $A \subseteq \mathbb{F}_2^n$ such that $V+A = \mathbb{F}_2^n$ uniquely (i.e., $v + a = v' + a'$ implies that $v=v'$ and $a=a'$, where $v,v' \in V$ and $a,a' \in A$). In some problems in coding theory and hashing we are given a putative tile $V$ and wish to know whether or not it is a tile. In this paper we give two computational criteria for certifying that $V$ is not a tile. The first involves the impossibility of a bin-packing problem, and the second involves the infeasibility of a linear program. We apply both criteria to a list of putative tiles given by Gordon, Miller, and Ostapenko [IEEE Trans. Inform. Theory, 56 (2010), pp. 984–991] in the context of hashing to find close matches, to show that none of them are, in fact, tiles.

On a Dispersion Problem in Grid Labeling

Minghui Jiang, Vincent Pilaud, and Pedro J. Tejada

SIAM J. Discrete Math. 26, pp. 39-51 (13 pages)

Online Publication Date: January 24, 2012

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Given $k$ labelings of a finite $d$-dimensional cubical grid, define the combined distance between two labels to be the sum of the $\ell_1$-distance between the two labels in each labeling. We want to construct $k$ labelings which maximize the minimum combined distance between any two labels. When $d=1$, this can be interpreted as placing $n$ nonattacking rooks in a $k$-dimensional chessboard of size $n$ in such a way to maximize the minimum $\ell_1$-distance between any two rooks. Rook placements are also known as Latin hypercube designs in the literature. In this paper, we revisit this problem with a more geometric approach. Instead of providing explicit but complicated formulas, we construct rook placements in a $k$-dimensional chessboard of size $n$ as certain lattice-like structures for certain well-chosen values of $n$. Then, we extend these constructions to any values of $n$ using geometric arguments. With this method, we present a clean and geometric description of the known optimal rook placements in the two-dimensional square grid. Furthermore, we provide asymptotically optimal constructions of $k$ labelings of $d$-dimensional cubical grids which maximize the minimum combined distance. Finally, we discuss the extension of this problem to labelings of an arbitrary graph. We prove that deciding whether a graph has two labelings with combined distance at least $3$ is at least as hard as graph isomorphism.

Labeling Planar Graphs without 4,5-Cycles with a Condition on Distance Two

Hai-Yang Zhu, Xin-Zhong Lu, Cui-Qi Wang, and Ming Chen

SIAM J. Discrete Math. 26, pp. 52-64 (13 pages)

Online Publication Date: January 26, 2012

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Wegner conjectured that for each planar graph $G$ with maximum degree $\Delta$ at least 4, $\chi(G^2)\leq\Delta+5$ if $4\leq\Delta\leq7$, and $\chi(G^2)\leq\lfloor \frac{3\Delta}{2}\rfloor +1$ if $\Delta\geq8$. Let $G$ be a planar graph without 4- and 5-cycles. In this paper, we discuss the $L(p,q)$-labeling of $G$ and show that $\lambda_{p,q}(G)\leq(2q-1)\Delta+6p+6q-6$ and $\lambda_{p,q}(G)\leq\max\{(2q-1)\Delta+6p+2q-4,9(2q-1)+8p-4,6(2q-1)+10p-5\},$ where $p$ and $q$ are positive integers with $p\geq q$. As a corollary, $\chi(G^2)\leq\Delta+7$ if $\Delta\leq7$, $\chi(G^2)\leq14$ if $\Delta=8$, and $\chi(G^2)\leq\Delta+5$ if $\Delta\geq9$.

Formulae for the Alon–Tarsi Conjecture

Douglas S. Stones

SIAM J. Discrete Math. 26, pp. 65-70 (6 pages)

Online Publication Date: January 26, 2012

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The sign of a Latin square is $-1$ if it has an odd number of rows and columns that are odd permutations; otherwise it is $+1$. Let $L^{\text{\scshape{even}}}_n$ and $L^{\text{\scshape{odd}}}_n$ be, respectively, the number of Latin squares of order $n$ with sign $+1$ and $-1$. The Alon–Tarsi conjecture asserts that $L^{\text{\scshape{even}}}_n \neq L^{\text{\scshape{odd}}}_n$ when $n$ is even. We prove that $L_n^{\text{\scshape{even}}}-L_n^{\text{\scshape{odd}}}=(-1)^{n(n-1)/2} \sum_{A \in B_n} (-1)^{\sigma_0(A)} \det(A)^n$, where $B_n$ is the set of $n \times n$ $\,(0,1)$-matrices and $\sigma_0(A)$ is the number of $0$ elements in $A$. We use this formula to give another proof of the Alon–Tarsi conjecture for $n=p-1$ for odd prime $p$.

Large $B_d$-Free and Union-free Subfamilies

János Barát, Zoltán Füredi, Ida Kantor, Younjin Kim, and Balázs Patkós

SIAM J. Discrete Math. 26, pp. 71-76 (6 pages)

Online Publication Date: February 09, 2012

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For a property $\Gamma$ and a family of sets ${\mathcal F}$, let $f({\mathcal F},\Gamma)$ be the size of the largest subfamily of ${\mathcal F}$ having property $\Gamma$. For a positive integer $m$, let $f(m,\Gamma)$ be the minimum of $f({\mathcal F},\Gamma)$ over all families of size $m$. A family ${\mathcal F}$ is said to be $B_d$-free if it has no subfamily ${\mathcal F}'=\{F_I: I \subseteq [d]\}$ of $2^d$ distinct sets such that for every $I,J \subseteq [d]$, both $F_I \cup F_J=F_{I \cup J}$ and $F_I \cap F_J = F_{I \cap J}$ hold. A family ${\mathcal F}$ is $a$-union-free if $F_1\cup \dots \cup F_a \neq F_{a+1}$ whenever $F_1,\dots,F_{a+1}$ are distinct sets in ${\mathcal F}$. We verify a conjecture of Erdős and Shelah that $f(m, B_2\text{\rm -free})=\Theta(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\text{\rm -free})$ and $f(m,a\text{\rm -union free})$.

Queue Layouts of Hypercubes

Petr Gregor, Riste Škrekovski, and Vida Vukašinović

SIAM J. Discrete Math. 26, pp. 77-88 (12 pages)

Online Publication Date: February 09, 2012

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A queue layout of a graph consists of a linear ordering $\sigma$ of its vertices and a partition of its edges into sets, called queues, such that in each set no two edges are nested with respect to $\sigma$. We show that the $n$-dimensional hypercube $Q_n$ has a layout into $n-\lfloor \log_2 n \rfloor$ queues for all $n\ge 1$. On the other hand, for every $\varepsilon>0$, every queue layout of $Q_n$ has more than $(\frac{1}{2}-\varepsilon) n-O(1/\varepsilon)$ queues and, in particular, more than $(n-2)/3$ queues. This improves previously known upper and lower bounds on the minimal number of queues in a queue layout of $Q_n$. For the lower bound we employ a new technique of out-in representations and contractions which may be of independent interest.

Cohen–Macaulay Graphs and Face Vectors of Flag Complexes

David Cook, II and Uwe Nagel

SIAM J. Discrete Math. 26, pp. 89-101 (13 pages)

Online Publication Date: February 09, 2012

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We introduce a construction on a flag complex that by means of modifying the associated graph generates a new flag complex whose $h$-vector is the face vector of the original complex. This construction yields a vertex-decomposable, hence Cohen–Macaulay, complex. From this we get a (nonnumerical) characterization of the face vectors of flag complexes and deduce also that the face vector of a flag complex is the $h$-vector of some vertex-decomposable flag complex. We conjecture that the converse of the latter is true and prove this, by means of an explicit construction, for $h$-vectors of Cohen–Macaulay flag complexes arising from bipartite graphs. We also give several new characterizations of bipartite graphs with Cohen–Macaulay or Buchsbaum independence complexes.

A Chain Theorem for $3^+$-Connected Graphs

Guoli Ding and Cheng Liu

SIAM J. Discrete Math. 26, pp. 102-113 (12 pages)

Online Publication Date: February 14, 2012

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A 3-connected graph is called $3^+$-connected if it has no 3-separation that separates a “large” fan or $K_{3,n}$ from the rest of the graph. It is proved in this paper that except for $K_4$, every $3^+$-connected graph has a $3^+$-connected proper minor that is at most two edges away from the original graph. This result is used to characterize $Q$-minor-free graphs, where $Q$ is obtained from the cube by contracting an edge.

Neighbor Systems, Jump Systems, and Bisubmodular Polyhedra

Akiyoshi Shioura

SIAM J. Discrete Math. 26, pp. 114-144 (31 pages)

Online Publication Date: February 16, 2012

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The concept of neighbor system, introduced by Hartvigsen in 2010, is a set of integral vectors satisfying a certain combinatorial property. In this paper, we reveal the relationship of neighbor systems with jump systems and with bisubmodular polyhedra. We first prove that for every neighbor system, there exists a jump system which has the same neighborhood structure as the original neighbor system. This shows that the concept of neighbor system is essentially equivalent to that of jump system. We next show that the convex closure of a neighbor system is an integral bisubmodular polyhedron. In addition, we give a characterization of neighbor systems using bisubmodular polyhedra. Finally, we consider the problem of minimizing a separable convex function on a neighbor system. It is shown that the problem can be solved in weakly polynomial time for a class of neighbor systems.

Griggs and Yeh's Conjecture and $L(p,1)$-labelings

Frédéric Havet, Bruce Reed, and Jean-Sébastien Sereni

SIAM J. Discrete Math. 26, pp. 145-168 (24 pages)

Online Publication Date: February 16, 2012

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An $L(p,1)$-labeling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geqslant p$ if dist$(x,y)=1$ and $|f(x)-f(y)|\geqslant 1$ if dist$(x,y)=2$, where dist$(x,y)$ is the distance between the two vertices $x$ and $y$ in the graph. The span of an $L(p,1)$-labeling $f$ is the difference between the largest and the smallest labels used by $f$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geqslant 2$ has an $L(2,1)$-labeling with span at most $\Delta^2$. We settle this conjecture for $\Delta$ sufficiently large. More generally, we show that for any positive integer $p$ there exists a constant $\Delta_p$ such that every graph with maximum degree $\Delta\geqslant \Delta_p$ has an $L(p,1)$-labeling with span at most $\Delta^2$. This yields that for each positive integer $p$, there is an integer $C_p$ such that every graph with maximum degree $\Delta$ has an $L(p,1)$-labeling with span at most $\Delta^2+C_p$.

Random Lifts of $K_5\backslashe$ are 3-Colorable

Babak Farzad and Dirk Oliver Theis

SIAM J. Discrete Math. 26, pp. 169-176 (8 pages)

Online Publication Date: February 16, 2012

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Amit, Linial, and Matoušek [Random Struct. Algorithms, 20 (2001), pp. 1–22] have raised the following question: Is the chromatic number of random $h$-lifts of $K_5$ asymptotically (for $h\to\infty$) almost surely (a.a.s.) equal to a single number? In this paper, we offer the following partial result: The chromatic number of a random lift of $K_5\backslash e$ is a.a.s. 3.

The Synchronizing Probability Function of an Automaton

Raphaël M. Jungers

SIAM J. Discrete Math. 26, pp. 177-192 (16 pages)

Online Publication Date: February 16, 2012

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We study the synchronization phenomenon for deterministic finite state automata and the related longstanding Černý conjecture. We formulate this conjecture in the setting of a two-player probabilistic game. Our goal is twofold. On the one hand, the probabilistic interpretation is of interest in its own right and can be applied to real-world situations. On the other hand, our formulation makes use of standard convex optimization techniques, which appear powerful to shed light on Černý's conjecture. We analyze the synchronization phenomenon through this particular point of view. Among other properties, we prove that the synchronization process cannot stagnate too long in a certain sense. We propose a new conjecture and demonstrate that its validity would imply Černý's conjecture. We show numerical evidence for the pertinence of the approach.

Roman Domination on 2-Connected Graphs

Chun-Hung Liu and Gerard J. Chang

SIAM J. Discrete Math. 26, pp. 193-205 (13 pages)

Online Publication Date: February 16, 2012

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A Roman dominating function of a graph $G$ is a function $f$$: V(G) \to \{0, 1, 2\}$ such that whenever $f(v)=0$, there exists a vertex $u$ adjacent to $v$ such that $f(u) = 2$. The weight of $f$ is $w(f) = \sum_{v \in V(G)} f(v)$. The Roman domination number $\gamma_R(G)$ of $G$ is the minimum weight of a Roman dominating function of $G$. Chambers, Kinnersley, Prince, and West [SIAM J. Discrete Math., 23 (2009), pp. 1575–1586] conjectured that $\gamma_R(G) \le \lceil 2n/3 \rceil$ for any $2$-connected graph $G$ of $n$ vertices. This paper gives counterexamples to the conjecture and proves that $\gamma_R(G) \le \max\{\lceil 2n/3 \rceil, 23n/34\}$ for any $2$-connected graph $G$ of $n$ vertices. We also characterize $2$-connected graphs $G$ for which $\gamma_R(G) = 23n/34$ when $23n/34 > \lceil 2n/3 \rceil$.

Nested Recurrence Relations with Conolly-like Solutions

Alejandro Erickson, Abraham Isgur, Bradley W. Jackson, Frank Ruskey, and Stephen M. Tanny

SIAM J. Discrete Math. 26, pp. 206-238 (33 pages)

Online Publication Date: February 23, 2012

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A nondecreasing sequence of positive integers is $(\alpha,\beta)$-Conolly, or Conolly-like for short, if for every positive integer $m$ the number of times that $m$ occurs in the sequence is $\alpha + \beta r_m$, where $r_m$ is $1$ plus the 2-adic valuation of $m$. A recurrence relation is $(\alpha, \beta)$-Conolly if it has an $(\alpha, \beta)$-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the form $A(n) = \sum_{i=1}^k A(n-s_i-\sum_{j=1}^{p_i} A(n-a_{ij}))$ with appropriate initial conditions. For any fixed integers $k$ and $p_1,p_2,\ldots, p_k$ we prove that there are only finitely many pairs $(\alpha, \beta)$ for which $A(n)$ can be $(\alpha, \beta)$-Conolly. For the case where $\alpha =0$ and $\beta =1$, we provide a bijective proof using labeled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence $H(n)=H(n-H(n-2)) + H(n-3-H(n-5))$ also has the Conolly sequence as a solution. When $k=2$ and $p_1=p_2$, we construct an example of an $(\alpha,\beta)$-Conolly recursion for every possible ($\alpha,\beta)$ pair, thereby providing the first examples of nested recursions with $p_i>1$ whose solutions are completely understood. Finally, in the case where $k=2$ and $p_1=p_2$, we provide an if and only if condition for a given nested recurrence $A(n)$ to be $(\alpha,0)$-Conolly by proving a very general ceiling function identity.

Nonextendible Latin Cuboids

Darryn Bryant, Nicholas J. Cavenagh, Barbara Maenhaut, Kyle Pula, and Ian M. Wanless

SIAM J. Discrete Math. 26, pp. 239-249 (11 pages)

Online Publication Date: February 23, 2012

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We show that for all integers $m \geqslant 4$ there exists a $2m\times 2m\times m$ latin cuboid that cannot be completed to a $2m\times 2m\times 2m$ latin cube. We also show that for all even $m>2$ there exists a $(2m{-}1)\times(2m{-}1)\times(m{-}1)$ latin cuboid that cannot be extended to any $(2m{-}1)\times(2m{-}1)\times m$ latin cuboid.

The MST of Symmetric Disk Graphs (in Arbitrary Metric Spaces) is Light

Shay Solomon

SIAM J. Discrete Math. 26, pp. 250-262 (13 pages)

Online Publication Date: February 28, 2012

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Consider an $n$-point metric space $M = (V,\delta)$ and a transmission range assignment $r: V \rightarrow \mathbb R^+$ that maps each point $v \in V$ to the disk of radius $r(v)$ around it. The symmetric disk graph (SDG) that corresponds to $M$ and $r$ is the undirected graph over $V$ whose edge set includes an edge $(u,v)$ if both $r(u)$ and $r(v)$ are no smaller than $\delta(u,v)$. SDGs are often used to model wireless communication networks. Abu-Affash et al. [Lecture Notes in Comput. Sci. 6139, Springer, Heidelberg, 2010, pp. 236–247] showed that for any $n$-point $2$-dimensional Euclidean space $M$, the weight of the minimum spanning tree (MST) of every connected SDG for $M$ is $O(\log n) \cdot w(MST(M))$, and that this bound is tight. However, the upper bound proof of Abu-Affash et al. relies heavily on basic geometric properties of constant-dimensional Euclidean spaces and does not extend to Euclidean spaces of super-constant dimension. A natural question that arises is whether this surprising upper bound of Abu-Affash et al. can be generalized for wider families of metric spaces, such as high-dimensional Euclidean spaces. In this paper we generalize the upper bound of Abu-Affash et al. for Euclidean spaces of any dimension. Furthermore, our upper bound extends to arbitrary metric spaces and, in particular, it applies to any of the normed spaces $\ell_p$. Specifically, we demonstrate that for any $n$-point metric space $M$, the weight of the MST of every connected SDG for $M$ is $O(\log n) \cdot w(MST(M))$.

Packing Squares with Profits

Klaus Jansen and Roberto Solis-Oba

SIAM J. Discrete Math. 26, pp. 263-279 (17 pages)

Online Publication Date: February 28, 2012

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We study the following square packing problem: Given a set $Q$ of squares with positive profits, the goal is to pack a subset of $Q$ into a rectangular bin $\mathcal R$ so that the total profit of the squares packed in $\mathcal R$ is maximized. Squares must be packed so that their sides are parallel to those of $\mathcal R$. We present a polynomial time approximation scheme for the problem, which for any value $\epsilon > 0$ finds and packs a subset $Q' \subseteq Q$ of profit at least $(1-\epsilon) OPT$, where $OPT$ is the profit of an optimum solution.

The Rigidity of Spherical Frameworks: Swapping Blocks and Holes

Wendy Finbow, Elissa Ross, and Walter Whiteley

SIAM J. Discrete Math. 26, pp. 280-304 (25 pages)

Online Publication Date: February 28, 2012

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A significant range of geometric structures whose rigidity has been explored, for both practical and theoretical purposes, are formed by modifying generically isostatic triangulated spheres. In block and hole structures $(\mathcal{P},\boldsymbol{p})$, some edges are removed (to make holes) and others are added (to create rigid subparts called blocks). Previous work noted a combinatorial analogy, in which blocks and holes played equivalent roles—so that they might be interchanged. In this paper, we geometrically connect stresses in such structures $(\mathcal{P},\boldsymbol{p})$ to first-order motions in a swapped structure $(\overline{\mathcal{P}},\boldsymbol{p})$—where holes become blocks and blocks become holes. When the initial structure is geometrically isostatic, this shows that the swapped structure is also geometrically isostatic, giving the strongest possible correspondence. We use an affine presentation of the statics and the motions to make the key underlying correspondences transparent.

Graphs That Admit Polyline Drawings with Few Crossing Angles

Eyal Ackerman, Radoslav Fulek, and Csaba D. Tóth

SIAM J. Discrete Math. 26, pp. 305-320 (16 pages)

Online Publication Date: March 06, 2012

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We consider graphs that admit polyline drawings where all crossings occur at the same angle $\alpha\in (0,\frac{\pi}{2}]$. We prove that every graph on $n$ vertices that admits such a polyline drawing with at most two bends per edge has $O(n)$ edges. This result remains true when each crossing occurs at an angle from a small set of angles. We also provide several extensions that might be of independent interest.

Families of Graph-different Hamilton Paths

János Körner, Silvia Messuti, and Gábor Simonyi

SIAM J. Discrete Math. 26, pp. 321-329 (9 pages)

Online Publication Date: March 13, 2012

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Let $\mathbb{D}\subseteq \mathbb{N}$ be an arbitrary subset of the natural numbers. For every $n$, let $M(n, \mathbb{D})$ be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph $K_n$ such that the union of any two paths from the family contains a not necessarily induced cycle of some length from $\mathbb{D}$. We determine or bound the asymptotics of $M(n, \mathbb{D})$ in various special cases. This problem is closely related to that of the permutation capacity of graphs and constitutes a further extension of the problem area around Shannon capacity. We also discuss how to generalize our cycle-difference problems and present an example where cycles are replaced by 4-cliques. These problems are in a natural duality to those of graph intersection, initiated by Erdős, Simonovits, and Sós. The lack of kernel structure as a natural candidate for optimum makes our problems quite challenging.

Reconstructing 3-Colored Grids from Horizontal and Vertical Projections is NP-Hard: A Solution to the 2-Atom Problem in Discrete Tomography

Christoph Dürr, Flavio Guiñez, and Martin Matamala

SIAM J. Discrete Math. 26, pp. 330-352 (23 pages)

Online Publication Date: March 13, 2012

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We consider the problem of coloring a grid using $k$ colors with the restriction that each row and each column has a specific number of cells of each color. This problem has been known as the $(k-1)$-atom problem in the discrete tomography community. In an already classical result, Ryser obtained a necessary and sufficient condition for the existence of such a coloring when two colors are considered. This characterization yields a linear time algorithm for constructing such a coloring when it exists. Gardner et al. showed that for $k\geqslant 7$ the problem is NP-hard. Afterward Chrobak and Dürr improved this result by proving that it remains NP-hard for $k\geqslant 4$. We close the gap by showing that for $k=3$ colors the problem is already NP-hard. In addition, we give some results on tiling tomography problems.

Universality of Random Graphs

Domingos Dellamonica,, Jr., Yoshiharu Kohayakawa, Vojtěch Rödl, and Andrzej Ruciński

SIAM J. Discrete Math. 26, pp. 353-374 (22 pages)

Online Publication Date: March 13, 2012

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We prove that asymptotically (as $n\to\infty$) almost all graphs with $n$ vertices and $C_dn^{2-\frac{1}{2d}} \log^{\frac{1}{d}} n$ edges are universal with respect to the family of all graphs with maximum degree bounded by $d$. Moreover, we provide an efficient deterministic embedding algorithm for finding copies of bounded degree graphs in graphs satisfying certain pseudorandom properties. We also prove a counterpart result for random bipartite graphs, where the threshold number of edges is even smaller but the embedding is randomized.

Constructing Constant Composition Codes via Distance-Increasing Mappings

Hsin-Lung Wu and Jen-Chun Chang

SIAM J. Discrete Math. 26, pp. 375-383 (9 pages)

Online Publication Date: March 13, 2012

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A distance-preserving mapping is a one-to-one function $f$ from $p$-ary vectors of length $m$ to $q$-ary vectors of length $n$ such that any two distinct $p$-ary vectors are mapped to two different $q$-ary vectors with an equal or greater Hamming distance. A distance-increasing mapping is a special distance-preserving mapping which strictly increases the distance by at least one if the distance of two distinct input vectors is less than the length of the output vectors. A constant composition code over a $k$-ary alphabet has the property that the numbers of occurrences of the $k$ symbols within a codeword are fixed for each codeword. One of the most important applications of distance-preserving mappings and distance-increasing mappings is to construct constant composition codes, of which the permutation codes are a special subclass. There are two results in this paper. First, we propose a swap-based distance-increasing mapping from binary vectors to quaternary constant composition vectors. Second, we prove that it is impossible to construct any swap-based distance-preserving mappings from binary vectors to ternary constant composition vectors under the swap model that we defined.

How to Choose the Best Twins

Bryn Garrod, Grzegorz Kubicki, and Michał Morayne

SIAM J. Discrete Math. 26, pp. 384-398 (15 pages)

Online Publication Date: March 15, 2012

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We consider a version of the secretary problem where each candidate has an identical twin. The aim, as in the classical problem, is to choose with the largest possible probability a top candidate, i.e., one of the best twins. We find an optimal stopping time for such a choice, the probability of success the optimal stopping time yields, and their asymptotic behavior.

Vertices Belonging to All Critical Sets of a Graph

Vadim E. Levit and Eugen Mandrescu

SIAM J. Discrete Math. 26, pp. 399-403 (5 pages)

Online Publication Date: March 20, 2012

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Let $G=(V,E)$ be a graph. A set $S\subseteq V$ is independent if no two vertices from $S$ are adjacent, while $\mathrm{core}(G)$ is the intersection of all maximum independent sets [V. E. Levit and E. Mandrescu, Discrete Appl. Math., 117 (2002), pp. 149–161]. The independence number $\alpha(G)$ is the cardinality of a largest independent set, and $\mu(G)$ is the size of a maximum matching of $G$. The neighborhood of $A\subseteq V$ is $\mathcal{N}(A)=\{v\in V:\mathcal{N}(v)\cap A\neq\emptyset\}$. The number $d_{c}(G)=\max\{\vert X\vert -\vert \mathcal{N}(X)\vert :X\subseteq V\}$ is called the critical difference of $G$, and $A$ is critical if $\vert A\vert -\vert \mathcal{N}% (A)\vert =d_{c}(G)$ [C. Q. Zhang, SIAM J. Discrete Math., 3 (1990), pp. 431–438]. We define $\mathrm{\ker}(G)$ as the intersection of all critical sets. In this paper we prove that if $d_{c}(G)\geq1$, then $\mathrm{\ker}(G)\subseteq\mathrm{core}(G)$ and $\vert \mathrm{\ker}(G)\vert >d_{c}(G) \geq\alpha(G) -\mu(G)$.

On the Complexity of MMSNP

Manuel Bodirsky, Hubie Chen, and Tomás Feder

SIAM J. Discrete Math. 26, pp. 404-414 (11 pages)

Online Publication Date: March 29, 2012

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Monotone monadic strict NP (MMSNP) is a class of computational problems that is closely related to the class of constraint satisfaction problems for constraint languages over finite domains. It is known that one of those classes has a complexity dichotomy if and only if the other class has. Whereas the dichotomy conjecture has been verified for several subclasses of constraint satisfaction problems, little is known about the the computational complexity for subclasses of MMSNP. In this paper we completely classify the complexity of MMSNP for the case where the obstructions are monochromatic and where loops in the input are forbidden. That is, we determine the computational complexity of natural partition problems of the following type. For fixed sets of finite structures ${\cal S}_1, \dots, {\cal S}_k$, decide whether a given loopless structure can be vertex-partitioned into $k$ parts such that for each $i \leq k$ none of the structures in ${\cal S}_i$ is homomorphic to the $i$th part.
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