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SIAM J. on Discrete Mathematics

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2010

Volume 24, Issue 4, pp. 1215-1762


Embedding into Bipartite Graphs

Julia Böttcher, Peter Heinig, and Anusch Taraz

SIAM J. Discrete Math. 24, pp. 1215-1233 (19 pages)

Online Publication Date: September 29, 2010

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The conjecture of Bollobás and Komlós, recently proved by Böttcher, Schacht, and Taraz [Math. Ann., 343 (2009), pp. 175–205], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac{1}{2}+\gamma)n$ when we have the additional structural information of the host graph $G$ being balanced bipartite. This complements results of Zhao [SIAM J. Discrete Math., 23 (2009), pp. 888–900], as well as Hladký and Schacht [SIAM J. Discrete Math., 24 (2010), pp. 357–362], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, our result can be used to prove that in every balanced bipartite graph $G$ on $2n$ vertices with minimum degree $(\frac{1}{2}+\gamma)n$ and $n$ sufficiently large, the set of Hamilton cycles of $G$ is a generating system for its cycle space.

On the Relative Generalized Hamming Weights of Linear Codes and their Subcodes

Zihui Liu, Jie Wang, and Xin-Wen Wu

SIAM J. Discrete Math. 24, pp. 1234-1241 (8 pages)

Online Publication Date: September 29, 2010

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We first present an equivalent definition of relative generalized Hamming weights of a linear code and its subcodes, and we develop a method using finite projective geometry. Making use of the equivalent definition and the projective-geometry method, all of the relative generalized Hamming weights of a 3-dimensional $q$-ary linear code and its subcodes will be determined.

A Lower Bound on the Transposition Diameter

Linyuan Lu and Yiting Yang

SIAM J. Discrete Math. 24, pp. 1242-1249 (8 pages)

Online Publication Date: September 29, 2010

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Sorting permutations by transpositions is an important and difficult problem in genome rearrangements. The transposition diameter $TD(n)$ is the maximum transposition distance among all pairs of permutations in $S_n$. It was previously conjectured [H. Eriksson et al., Discrete Math., 241 (2001), pp. 289–300] that $TD(n)\leq\lceil\frac{n+1}{2}\rceil$. This conjecture was disproved by Elias and Hartman [IEEE/ACM Trans. Comput. Biol. Bioinform., 3 (2006), pp. 369–379] by showing $TD(n)\geq\lfloor\frac{n+1}{2}\rfloor+1$. In this paper we improved the lower bound to $TD(n)\geq\frac{17}{33}n+\frac{1}{33}$ via computation.

Hopf Structures on the Multiplihedra

Stefan Forcey, Aaron Lauve, and Frank Sottile

SIAM J. Discrete Math. 24, pp. 1250-1271 (22 pages)

Online Publication Date: September 29, 2010

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We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Most compelling among these are two distinct structures of a Hopf module over the Loday–Ronco Hopf algebra.

Mixed Statistics on 01-Fillings of Moon Polyominoes

William Y. C. Chen, Andrew Y. Z. Wang, Catherine H. Yan, and Alina F. Y. Zhao

SIAM J. Discrete Math. 24, pp. 1272-1290 (19 pages)

Online Publication Date: October 05, 2010

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We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of 01-fillings of moon polyominoes. Let $\mathcal{M}$ be a moon polyomino with $n$ rows and $m$ columns. Consider all the 01-fillings of $\mathcal{M}$ in which every row has at most one 1. We introduce four mixed statistics with respect to a bipartition of rows or columns of $\mathcal{M}$. More precisely, let $S\subseteq\{1,2,\dots,n\}$ and let $\mathcal{R}(S)$ be the union of rows whose indices are in $S$. For any filling $M$, the top-mixed (resp., bottom-mixed) statistic $\alpha(S;M)$ (resp., $\beta(S;M)$) is the sum of the number of northeast chains whose top (resp., bottom) cell is in $\mathcal{R}(S)$, together with the number of southeast chains whose top (resp., bottom) cell is in the complement of $\mathcal{R}(S)$. Similarly, we define the left-mixed and right-mixed statistics $\gamma(T;M)$ and $\delta(T;M)$, where $T$ is a subset of the column index set $\{1,2,\dots,m\}$. Let $\lambda(A;M)$ be any of these four statistics $\alpha(S;M)$, $\beta(S;M)$, $\gamma(T;M)$, and $\delta(T;M)$; we show that the joint distribution of the pair $(\lambda(A;M),\lambda(\bar{A};M))$ is symmetric and independent of the subsets $S,T$. In particular, the pair of statistics $(\lambda(A;M),\lambda(\bar{A};M))$ is equidistributed with $(\mathrm{se}(M),\mathrm{ne}(M))$, where $\mathrm{se}(M)$ and $\mathrm{ne}(M)$ are the numbers of southeast chains and northeast chains of $M$, respectively.

Symmetry as a Sufficient Condition for a Finite Flex

Bernd Schulze

SIAM J. Discrete Math. 24, pp. 1291-1312 (22 pages)

Online Publication Date: October 05, 2010

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We show that if the joints of a bar and joint framework $(G,p)$ are positioned as “generically” as possible subject to given symmetry constraints and $(G,p)$ possesses a “fully symmetric” infinitesimal flex (i.e., the velocity vectors of the infinitesimal flex remain unaltered under all symmetry operations of $(G,p)$), then $(G,p)$ also possesses a finite flex which preserves the symmetry of $(G,p)$ throughout the path. This and other related results are obtained by symmetrizing techniques described by L. Asimov and B. Roth in their 1978 paper “The Rigidity of Graphs” [Trans. Amer. Math. Soc., 245 (1978), pp. 279–289] and by using the fact that the rigidity matrix of a symmetric framework can be transformed into a block-diagonalized form by means of group representation theory. The finite flexes that can be detected with these symmetry-based methods can in general not be found with the analogous nonsymmetric methods.

Complete Minors and Independence Number

Jacob Fox

SIAM J. Discrete Math. 24, pp. 1313-1321 (9 pages)

Online Publication Date: October 12, 2010

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Let $G$ be a graph with $n$ vertices and independence number $\alpha$. Hadwiger's conjecture implies that $G$ contains a clique minor of order at least $n/\alpha$. In 1982, Duchet and Meyniel proved that this bound holds within a factor 2. Our main result gives the first improvement on their bound by an absolute constant factor. We show that $G$ contains a clique minor of order larger than $.504n/\alpha$. We also prove related results giving lower bounds on the order of the largest clique minor.

Surviving Rates of Graphs with Bounded Treewidth for the Firefighter Problem

Leizhen Cai, Yongxi Cheng, Elad Verbin, and Yuan Zhou

SIAM J. Discrete Math. 24, pp. 1322-1335 (14 pages)

Online Publication Date: October 12, 2010

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The firefighter problem is the following discrete-time game on a graph. Initially, a fire starts at a vertex of the graph. In each round, a firefighter protects one vertex not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. The surviving rate of a graph is the average percentage of vertices that can be saved when a fire starts randomly at one vertex of the graph, which measures the defense ability of a graph as a whole. In this paper, we study the surviving rates of graphs with bounded treewidth. We prove that the surviving rate of every $n$-vertex outerplanar graph is at least $1-\Theta(\frac{\log n}{n})$, which is asymptotically tight. We also prove that if $k$ firefighters are available in each round, then the surviving rate of an $n$-vertex graph with treewidth at most $k$ is at least $1-O(\frac{k^{2}\log n}{n})$. Furthermore, we show that the greedy strategy of Hartnell and Li [Congr. Numer., 145 (2000), pp. 187–192] for trees saves at least $1-\Theta(\frac{\log n}{n})$ percent of vertices on average for an $n$-vertex tree. Our results settle a conjecture and two problems of Cai and Wang [SIAM J. Discrete Math., 23 (2009), pp. 1814–1826] in affirmative.

Strong Transversals in Hypergraphs and Double Total Domination in Graphs

Michael A. Henning and Anders Yeo

SIAM J. Discrete Math. 24, pp. 1336-1355 (20 pages)

Online Publication Date: October 14, 2010

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Let $H$ be a 3-uniform hypergraph of order $n$ and size $m$, and let $T$ be a subset of vertices of $H$. The set $T$ is a strong transversal in $H$ if $T$ contains at least two vertices from every edge of $H$. The strong transversal number $\tau_s(H)$ of $H$ is the minimum size of a strong transversal in $H$. We show that $7\tau_s(H)\leq4n+2m$, and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph $H$ of order $n\geq6$ and size $m$ that achieves equality in this bound. A set $S$ of vertices in a graph $G$ is a double total dominating set of $G$ if every vertex of $G$ is adjacent to at least two vertices in $S$. The minimum cardinality of a double total dominating set of $G$ is the double total domination number $\gamma_{\times2,t}(G)$ of $G$. Let $G$ be a connected graph of order $n$ with minimum degree at least three. As an application of our hypergraph results, we show that $\gamma_{\times2,t}(G)\leq6n/7$ with equality if and only if $G$ is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if $G$ is not the Heawood graph, we show that $\gamma_{\times2,t}(G)\leq11n/13$, while if $G$ is a cubic graph different from the Heawood graph, we show that $\gamma_{\times2,t}(G)\leq5n/6$, and this bound is sharp.

Abacus Proofs of Schur Function Identities

Nicholas A. Loehr

SIAM J. Discrete Math. 24, pp. 1356-1370 (15 pages)

Online Publication Date: October 14, 2010

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This article uses combinatorial objects called labeled abaci to give direct combinatorial proofs of many familiar facts about Schur polynomials. We use abaci to prove the Pieri rules, the Littlewood–Richardson rule, the equivalence of the tableau definition and the determinant definition of Schur polynomials, and the combinatorial interpretation of the inverse Kostka matrix (first given by Eğecioğlu and Remmel). The basic idea is to regard formulas involving Schur polynomials as encoding bead motions on abaci. The proofs of the results just mentioned all turn out to be manifestations of a single underlying theme: when beads bump, objects cancel.

The Tradeoff Function for a Class of $\mathrm{RLL}(d,k)$ Constraints

Erez Louidor

SIAM J. Discrete Math. 24, pp. 1371-1398 (28 pages)

Online Publication Date: October 28, 2010

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A reverse concatenation coding scheme for storage systems in which the information is encoded first by a modulation (constraint) code and then by a systematic error-correcting code is considered. In this scheme, the output of the modulation coding stage has certain positions left “unconstrained” in the sense that any way of filling them with bits results in a sequence that satisfies the constraint. These positions are then used to store the parity-check bits of the error-correcting code so that the result is a valid constrained sequence. The tradeoff function defines the maximum overall rate of the encoding scheme for a given density of unconstrained positions. This function is determined for two families of run length limited (RLL) constraints: $\mathrm{RLL}(d,\infty)$ and $\mathrm{RLL}(d,2d+2)$. For $\mathrm{RLL}(d,2d+2)$, a curious dichotomy in the shape of the tradeoff function between different ranges of values of $d$ is shown to exist.

Combinatorics and Geometry of Finite and Infinite Squaregraphs

Hans-Jürgen Bandelt, Victor Chepoi, and David Eppstein

SIAM J. Discrete Math. 24, pp. 1399-1440 (42 pages)

Online Publication Date: November 04, 2010

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Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic one and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees, and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that minimum-size median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard. Finite squaregraphs can be recognized in linear time by a Breadth-First-Search.

Approximation of Partial Capacitated Vertex Cover

Reuven Bar-Yehuda, Guy Flysher, Julián Mestre, and Dror Rawitz

SIAM J. Discrete Math. 24, pp. 1441-1469 (29 pages)

Online Publication Date: November 04, 2010

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We study the partial capacitated vertex cover problem (PCVC) in which the input consists of a graph $G$ and a covering requirement $L$. Each edge $e$ in $G$ is associated with a demand (or load) $\ell(e)$, and each vertex $v$ is associated with a (soft) capacity $c(v)$ and a weight $w(v)$. A feasible solution is an assignment of edges to vertices such that the total demand of assigned edges is at least $L$. The weight of a solution is $\sum_{v}\alpha(v)w(v)$, where $\alpha(v)$ is the number of copies of $v$ required to cover the demand of the edges that are assigned to $v$. The goal is to find a solution of minimum weight. We consider three variants of PCVC. In PCVC with separable demands the only requirement is that the total demand of edges assigned to $v$ is at most $\alpha(v)c(v)$. In PCVC with inseparable demands there is an additional requirement that if an edge is assigned to $v$, then it must be assigned to one of its copies. The third variant is the unit demands version. We present 3-approximation algorithms for both PCVC with separable demands and PCVC with inseparable demands. We also present a 2-approximation algorithm for PCVC with unit demands. We show that similar results can be obtained for PCVC in hypergraphs and for the prize collecting version of capacitated vertex cover. Our algorithms are based on a unified approach for designing and analyzing approximation algorithms for capacitated covering problems. This approach yields simple algorithms whose analyses rely on the local ratio technique and sophisticated charging schemes.

Edge-Connectivity, Eigenvalues, and Matchings in Regular Graphs

Suil O and Sebastian M. Cioabă

SIAM J. Discrete Math. 24, pp. 1470-1481 (12 pages)

Online Publication Date: November 04, 2010

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In this paper, we study the relationship between eigenvalues and the existence of certain subgraphs in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a $t$-edge-connected $d$-regular graph when $t\leq d-2$. This work extends some classical results of von Baebler [Comment. Math. Helv., 10 (1937), pp. 275–287] and Berge [Théorie des Graphes et Ses Applications, Collection Universitaire de Mathematiques II, Dunod, Paris, 1958] and more recent work of Cioabă, Gregory, and Haemers [J. Combin. Theory Ser. B, 99 (2009), pp. 287–297]. We also study the relationships between the eigenvalues of a $d$-regular $t$-edge-connected graph $G$ and the maximum number of pairwise disjoint connected subgraphs in $G$ that are each joined to the rest of the graph by exactly $t$ edges.

Routing Numbers of Cycles, Complete Bipartite Graphs, and Hypercubes

Wei-Tian Li, Linyuan Lu, and Yiting Yang

SIAM J. Discrete Math. 24, pp. 1482-1494 (13 pages)

Online Publication Date: November 04, 2010

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The routing number $rt(G)$ of a connected graph $G$ is the minimum integer $r$ so that every permutation of vertices can be routed in $r$ steps by swapping the ends of disjoint edges. In this paper, we study the routing numbers of cycles, complete bipartite graphs, and hypercubes. We prove that $rt(C_n)=n-1$ (for $n\geq3$) and for $s\geq t$, $rt(K_{s,t})=\lfloor\frac{3s}{2t}\rfloor+O(1)$. We also prove $n+1\leq rt(Q_n)\leq2n-2$ for $n\geq3$. The lower bound $rt(Q_n)\geq n+1$ was previously conjectured by Alon, Chung, and Graham [SIAM J. Discrete Math., 7 (1994), pp. 513–530]. A variation, called fractional routing number, is also considered in this paper.

Embedding Spanning Trees in Random Graphs

Michael Krivelevich

SIAM J. Discrete Math. 24, pp. 1495-1500 (6 pages) | Cited 1 time

Online Publication Date: November 11, 2010

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We prove that if $T$ is a tree on $n$ vertices with maximum degree $\Delta$ and the edge probability $p(n)$ satisfies $np\geq C\max\{\Delta\log n,n^{\epsilon}\}$ for some constant $\epsilon>0$, then with high probability the random graph $G(n,p)$ contains a copy of $T$. The obtained bound on the edge probability is shown to be essentially tight for $\Delta=n^{\Theta(1)}$.

Large-Girth Roots of Graphs

Anna Adamaszek and MichaŁ Adamaszek

SIAM J. Discrete Math. 24, pp. 1501-1514 (14 pages)

Online Publication Date: November 11, 2010

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We study the problem of recognizing graph powers and computing roots of graphs. Our focus is on classes of graphs with no short cycles. We provide a polynomial time recognition algorithm for $r$-th powers of graphs of girth at least $2r+3$, thus improving a recently conjectured bound. Our algorithm also finds all $r$-th roots of a given graph that have girth at least $2r+3$ and no degree one vertices, which is a step toward a recent conjecture of Levenshtein [Discrete Math., 308 (2008), pp. 993–998] that such roots should be unique. Similar algorithms have so far been designed only for $r=2,3$. On the negative side, we prove that recognition of graph powers becomes an NP-complete problem when the bound on girth is about twice smaller. (Anna Adamaszek's correct affiliation is Department of Computer Science and Centre for Discrete Mathematics and Its Applications (DIMAP), University of Warwick, Coventry, CV4 7AL, UK.)

Multiple Coloring of Cone Graphs

Zhishi Pan and Xuding Zhu

SIAM J. Discrete Math. 24, pp. 1515-1526 (12 pages)

Online Publication Date: November 30, 2010

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A $k$-fold coloring of a graph assigns to each vertex a set of $k$ colors, and color sets assigned to adjacent vertices are disjoint. The $k$th chromatic number $\chi_k(G)$ of a graph $G$ is the minimum total number of colors needed in a $k$-fold coloring of $G$. Given a graph $G=(V,E)$ and an integer $m\geq0$, the $m$-cone of $G$, denoted by $\mu_m(G)$, has vertex set $(V\times\{0,1,\dots,m\})\cup\{u\}$ in which $u$ is adjacent to every vertex of $V\times\{m\}$, and $(x,i)(y,j)$ is an edge if $xy\in E$ and $i=j=0$ or $xy\in E$ and $|i-j|=1$. This paper studies the $k$th chromatic number of the cone graphs. An upper bound for $\chi_k(\mu_m(G))$ in terms of $\chi_k(G)$, $k$, and $m$ are given. In particular, it is proved that for any graph $G$, if $m\geq2k$, then $\chi_k(\mu_m(G))\leq\chi_k(G)+1$. We also find a surprising connection between the $k$th chromatic number of the cone graph of $G$ and the circular chromatic number of $G$. It is proved that if $\chi_k(G)/k>\chi_c(G)$ and $\chi_k(G)$ is even, then for sufficiently large $m$, $\chi_k(\mu_m(G))=\chi_k(G)$. In particular, if $\chi(G)>\chi_c(G)$ and $\chi(G)$ is even, then for sufficiently large $m$, $\chi(\mu_m(G))=\chi(G)$.

Re-embeddings of Maximum 1-Planar Graphs

Yusuke Suzuki

SIAM J. Discrete Math. 24, pp. 1527-1540 (14 pages)

Online Publication Date: November 30, 2010

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In this paper, we examine the re-embeddability of maximum 1-planar graphs. In particular, we prove that every optimal 1-planar graph is uniquely 1-embeddable on the sphere except for a sequence of graphs that are minimal with respect to certain reductions. These optimal 1-planar graphs are closely related to their quadrangular subgraphs. We also give a generating theorem for optimal 1-planar graphs.

Designing Steiner Networks with Unicyclic Connected Components: An Easy Problem

Walid Ben-Ameur and Makhlouf Hadji

SIAM J. Discrete Math. 24, pp. 1541-1557 (17 pages)

Online Publication Date: November 30, 2010

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This paper focuses on the design of minimum-cost networks satisfying two technical constraints. First, the connected components should be unicyclic. Second, some given special nodes must belong to cycles. This problem is a generalization of two known problems: the perfect binary 2-matching problem and the problem of computing a minimum-weight basis of the bicircular matroid. It turns out that the problem is polynomially solvable. An exact extended linear formulation is provided. We also present a partial description of the convex hull of the incidence vectors of these Steiner networks. Polynomial-time separation algorithms are described. One of them is a generalization of the Padberg–Rao algorithm to separate blossom inequalities.

The Limit Shape of Large Alternating Sign Matrices

F. Colomo and A. G. Pronko

SIAM J. Discrete Math. 24, pp. 1558-1571 (14 pages)

Online Publication Date: November 30, 2010

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The problem of the limit shape of large alternating sign matrices (ASMs) is addressed by studying the emptiness formation probability (EFP) in the domain wall six-vertex model. Assuming that the limit shape arises in correspondence with the “condensation” of almost all solutions of the saddle-point equations for certain multiple integral representations for EFP, a conjectural expression for the limit shape of large ASMs is derived. The case of $3$-enumerated ASMs is also considered.

Equitable Coloring of Sparse Planar Graphs

Rong Luo, Jean-Sébastien Sereni, D. Christopher Stephens, and Gexin Yu

SIAM J. Discrete Math. 24, pp. 1572-1583 (12 pages)

Online Publication Date: November 30, 2010

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A proper vertex coloring of a graph $G$ is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold $\chi_{eq}^*(G)$ of $G$ is the smallest integer $m$ such that $G$ is equitably $n$-colorable for all $n\geq m$. We show that for planar graphs $G$ with minimum degree at least two, $\chi_{eq}^*(G)\leq4$ if the girth of $G$ is at least 10, and $\chi_{eq}^*(G)\leq3$ if the girth of $G$ is at least 14.

Ends and Vertices of Small Degree in Infinite Minimally $k$-(Edge)-Connected Graphs

Maya Stein

SIAM J. Discrete Math. 24, pp. 1584-1596 (13 pages)

Online Publication Date: December 07, 2010

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Bounds on the minimum degree and on the number of vertices attaining it have been much studied for finite edge-/vertex-minimally $k$-connected/$k$-edge-connected graphs. We give an overview of the results known for finite graphs and show that most of these carry over to infinite graphs if we consider ends of small degree as well as vertices.

The Balanced Decomposition Number and Vertex Connectivity

Shinya Fujita and Henry Liu

SIAM J. Discrete Math. 24, pp. 1597-1616 (20 pages)

Online Publication Date: December 07, 2010

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The balanced decomposition number $f(G)$ of a graph $G$ was introduced by Fujita and Nakamigawa [Discr. Appl. Math., 156 (2008), pp. 3339–3344]. A balanced coloring of a graph $G$ is a coloring of some of the vertices of $G$ with two colors, such that there is the same number of vertices in each color. Then, $f(G)$ is the minimum integer $s$ with the following property: For any balanced coloring of $G$, there is a partition $V(G)=V_1\,\dot\cup\,\cdots\,\dot\cup\,V_r$ such that, for every $i$, $V_i$ induces a connected subgraph, $|V_i|\leq s$, and $V_i$ contains the same number of colored vertices in each color. Fujita and Nakamigawa studied the function $f(G)$ for many basic families of graphs, and demonstrated some applications. In this paper, we shall continue the study of the function $f(G)$. We give a characterization for noncomplete graphs $G$ of order $n$ which are $\lfloor\frac{n}{2}\rfloor$-connected, in view of the balanced decomposition number. We shall prove that a necessary and sufficient condition for such $\lfloor\frac{n}{2}\rfloor$-connected graphs $G$ is $f(G)=3$. We shall also determine $f(G)$ when $G$ is a complete multipartite graph, and when $G$ is a generalized $\Theta$-graph (i.e., a graph which is a subdivision of a multiple edge). Some applications will also be discussed. Further results about the balanced decomposition number also appear in two subsequent papers by Fujita and Liu.

On the Number of Two-Dimensional Threshold Functions

Max A. Alekseyev

SIAM J. Discrete Math. 24, pp. 1617-1631 (15 pages) | Cited 1 time

Online Publication Date: December 07, 2010

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A two-dimensional threshold function of $k$-valued logic can be viewed as a coloring of the points of a $k\times k$ square lattice into two colors such that there exists a straight line separating points of different colors. For the number of such functions only asymptotic bounds are known. We give an exact formula for the number of two-dimensional threshold functions and derive more accurate asymptotics.

Thomassen's Choosability Argument Revisited

David R. Wood and Svante Linusson

SIAM J. Discrete Math. 24, pp. 1632-1637 (6 pages)

Online Publication Date: December 07, 2010

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Thomassen (J. Combin. Theory Ser. B, 62 (1994), pp. 180–181) proved that every planar graph is 5-choosable. This result was generalized by Škrekovski (Discrete Math., 190 (1998), pp. 223–226) and He, Miao, and Shen (Discrete Math., 308 (2008), pp. 4024–4026), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterization of $K_5$-minor-free graphs due to Wagner (Math. Ann., 114 (1937), pp. 570–590). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture.

Neighbor Systems and the Greedy Algorithm

David Hartvigsen

SIAM J. Discrete Math. 24, pp. 1638-1661 (24 pages) | Cited 1 time

Online Publication Date: December 07, 2010

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A neighbor system, introduced in this paper, is a collection of integral vectors in $\mathbb{R}^{n}$ with some special structure. Such collections (slightly) generalize jump systems, which, in turn, generalize integral bisubmodular polyhedra, integral polymatroids, delta-matroids, matroids, and other structures. We show that neighbor systems provide a systematic and simple way to characterize these structures. One of our main results is a simple greedy algorithm for optimizing over (finite) neighbor systems starting from any feasible vector. The algorithm is (essentially) identical to the usual greedy algorithm on matroids and integral polymatroids when the starting vector is zero. But in all other cases, from matroids through jump systems, it appears to be a new greedy algorithm. We end the paper by introducing another structure, which is more general than neighbor systems, and indicate that essentially the same greedy algorithm also works for this structure.

A More Relaxed Model for Graph-Based Data Clustering: $s$-Plex Cluster Editing

Jiong Guo, Christian Komusiewicz, Rolf Niedermeier, and Johannes Uhlmann

SIAM J. Discrete Math. 24, pp. 1662-1683 (22 pages)

Online Publication Date: December 08, 2010

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We introduce the $s$-Plex Cluster Editing problem as a generalization of the well-studied Cluster Editing problem; both are NP-hard and both are motivated by graph-based data clustering. Instead of transforming a given graph by a minimum number of edge modifications into a disjoint union of cliques (this is Cluster Editing), the task in the case of $s$-Plex Cluster Editing is to transform a graph into a cluster graph consisting of a disjoint union of so-called $s$-plexes. Herein, an $s$-plex is a vertex set $S$ inducing a subgraph in which every vertex has degree at least $|S|-s$. Cliques are 1-plexes. The advantage of $s$-plexes for $s\geq2$ is that they allow us to model a more relaxed cluster notion ($s$-plexes instead of cliques), better reflecting inaccuracies of the input data. We develop a provably effective preprocessing based on data reduction (yielding a so-called problem kernel), a forbidden subgraph characterization of $s$-plex cluster graphs, and a depth-bounded search tree which is used to find optimal edge modification sets. Altogether, this yields efficient algorithms in case of moderate numbers of edge modifications; this is often a reasonable assumption under a maximum parsimony model for data clustering.

On Additive Doubling and Energy

Nets Hawk Katz and Paul Koester

SIAM J. Discrete Math. 24, pp. 1684-1693 (10 pages)

Online Publication Date: December 16, 2010

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We discuss some ideas related to the polynomial Freiman–Ruzsa conjecture. We show that there is a universal $\epsilon>0$ so that any subset of an abelian group with subtractive doubling $K$ must be polynomially related to a set with additive energy at least $\frac{1}{K^{1-\epsilon}}$. This means that the main difficulty in proving the polynomial Freiman–Ruzsa conjecture consists of studying sets whose energy is greater than that implied by their doubling. One example is a generalized arithmetic progression of high dimension which cannot occur in the finite characteristic setting.

Flooding Time of Edge-Markovian Evolving Graphs

Andrea E. F. Clementi, Claudio Macci, Angelo Monti, Francesco Pasquale, and Riccardo Silvestri

SIAM J. Discrete Math. 24, pp. 1694-1712 (19 pages)

Online Publication Date: December 16, 2010

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=1We introduce stochastic time-dependency in evolving graphs: starting from an initial graph, at every time step, every edge changes its state (existing or not) according to a two-state Markovian process with probabilities $p$ (edge birth-rate) and $q$ (edge death-rate). If an edge exists at time $t$, then, at time $t+1$, it dies with probability $q$. If instead the edge does not exist at time $t$, then it will come into existence at time $t+1$ with probability $p$. Such an evolving graph model is a wide generalization of time-independent dynamic random graphs [A. E. F. Clementi, A. Monti, F. Pasquale, and R. Silvestri, J. Comput. System Sci., 75 (2009), pp. 213–220] and will be called edge-Markovian evolving graphs. We investigate the speed of information spreading in such evolving graphs. We provide nearly tight bounds (which in fact turn out to be tight for a wide range of probabilities $p$ and $q$) on the completion time of the flooding mechanism aiming to broadcast a piece of information from a source node to all nodes. In particular, we provide i) a tight characterization of the class of edge-Markovian evolving graphs where flooding time is constant and, thus, it does not asymptotically depend on the initial graph; ii) a tight characterization of the class of edge-Markovian evolving graphs where flooding time does not asymptotically depend on the edge death-rate $q$. An interesting consequence of our results is that information spreading can be fast even if the graph, at every time step, is very sparse and disconnected. Furthermore, our bounds imply that the flooding time can be exponentially shorter than the mixing time of the edge-Markovian graph.

Obnoxious Centers in Graphs

Sergio Cabello and Günter Rote

SIAM J. Discrete Math. 24, pp. 1713-1730 (18 pages)

Online Publication Date: December 16, 2010

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We consider the problem of finding obnoxious centers in graphs. For arbitrary graphs with $n$ vertices and $m$ edges, we give a randomized algorithm with $O(n\log^{2}n+m\log n)$ expected time. For planar graphs, we give algorithms with $O(n\log n)$ expected time and $O(n\log^{3}n)$ worst-case time. For graphs with bounded treewidth, we give an algorithm taking $O(n\log n)$ worst-case time. The algorithms make use of parametric search and several results for computing distances on graphs of bounded treewidth and planar graphs.

Bridges in Highly Connected Graphs

Paul Wollan

SIAM J. Discrete Math. 24, pp. 1731-1741 (11 pages)

Online Publication Date: December 21, 2010

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Let $\mathcal{P}=\{P_1,\dots,P_l\}$ be a set of internally disjoint paths contained in a graph $G$, and let $S$ be the subgraph defined by $\bigcup_{i=1}^{t}P_i$. A $\mathcal{P}$-bridge is either an edge of $G-E(S)$ with both endpoints in $V(S)$ or a component $C$ of $G-V(S)$ along with all the edges from $V(C)$ to $V(S)$. The attachments of a bridge $B$ are the vertices of $V(B)\cap V(S)$. A bridge $B$ is $k$-stable if there does not exist a subset of at most $k-1$ paths in $\mathcal{P}$ containing every attachment of $B$. A classic theorem of Tutte [Graph Theory, Addison–Wesley, Menlo Park, CA, 1984] states that if $G$ is a 3-connected graph, there exists a set of internally disjoint paths $\mathcal{P}'=\{P_1',\dots,P_l'\}$ such that $P_i$ and $P_i'$ have the same endpoints for $1\leq i\leq t$ and every $\mathcal{P}'$-bridge is 2-stable. We prove that if the graph is sufficiently connected, the paths $P_1',\dots,P_l'$ may be chosen so that every bridge containing at least two edges is, in fact, $k$-stable. We also give several simple applications of this theorem related to a conjecture of Lovász [Problems in Graph Theory, Recent Advances in Graph Theory, M. Felder, ed., Acadamia, Prague, 1975] on deleting paths while maintaining high connectivity.

A Construction of Infinite Sets of Intertwines for Pairs of Matroids

Joseph E. Bonin

SIAM J. Discrete Math. 24, pp. 1742-1752 (11 pages)

Online Publication Date: December 21, 2010

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An intertwine of a pair of matroids is a matroid such that it, but none of its proper minors, has minors that are isomorphic to each matroid in the pair. For pairs for which neither matroid can be obtained, up to isomorphism, from the other by taking free extensions, free coextensions, and minors, we construct a family of rank-$k$ intertwines for each sufficiently large integer $k$. We also treat some properties of these intertwines.

Bessel Polynomials and the Partial Sums of the Exponential Series

Ömer Eğecioğlu

SIAM J. Discrete Math. 24, pp. 1753-1762 (10 pages)

Online Publication Date: December 21, 2010

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Let $e_k(x)$ denote the $k$-th partial sum of the Maclaurin series for the exponential function. Define the $(n+1)\times(n+1)$ Hankel determinant by setting $\widetilde{H}_n(x)=\det[e_{i+j}(x)]_{0\leq i,j\leq n}$. We give a closed form evaluation of this determinant in terms of the Bessel polynomials using the method of recently introduced $\gamma$-operators.
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