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SIAM J. on Discrete Mathematics

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2012

Volume 26, Issue 1 (partial)


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 $23n/34 > \lceil 2n/3 \rceil$.
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