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SIAM J. on Discrete Mathematics

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2008

Volume 22, Issue 2, pp. 425-847


Asymptotic Determination of Edge-Bandwidth of Multidimensional Grids and Hamming Graphs

Reza Akhtar, Tao Jiang, and Zevi Miller

SIAM J. Discrete Math. 22, pp. 425-449 (25 pages)

Online Publication Date: March 20, 2008

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The edge-bandwidth $B'(G)$ of a graph $G$ is the bandwidth of the line graph of $G$. More specifically, for any bijection $f: E(G)\to \{1,2,\ldots, |E(G)|\}$, let $B'(f,G)=\max\{|f(e_1)- f(e_2)|: \mbox{$e1$ and $e2$ are incident edges of $G$}\}$, and let $B'(G)=\min_f B'(f,G)$. We determine asymptotically the edge-bandwidth of $d$-dimensional grids $P_n^d$ and of the Hamming graph $K_n^d$, the $d$-fold Cartesian product of $K_n$. Our results are as follows. (i) For fixed $d$ and $n\to \infty$, $B'(P_n^d)=c(d)d n^{d-1}+O(n^{d-{3\over2}})$, where $c(d)$ is a constant depending on $d$, which we determine explicitly. (ii) For fixed even $n$ and $d\to \infty$, $B'(K_n^d)=(1+o(1))\sqrt{d\over {2\pi}}\, n^d (n-1)$. Our results extend recent results by Balogh, Mubayi, and Pluhár [Theoret. Comput. Sci., 359 (2006), pp. 43–57], who determined $B'(P_n^2)$ asymptotically as a function of $n$ and $B'(K_2^d)$ asymptotically as a function of $d$.

Polychromatic Colorings of Subcubes of the Hypercube

David Offner

SIAM J. Discrete Math. 22, pp. 450-454 (5 pages)

Online Publication Date: March 20, 2008

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Alon, Krech, and Szabó [SIAM J. Discrete Math., 21 (2007), pp. 66–72] called an edge-coloring of a hypercube with $p$ colors such that every subcube of dimension $d$ contains every color a $d$-polychromatic $p$-coloring. Denote by $p_d$ the maximum number of colors with which it is possible to $d$-polychromatically color any hypercube. We find the exact value of $p_d$ for all values of $d$.

Matroid Complexity and Nonsuccinct Descriptions

Dillon Mayhew

SIAM J. Discrete Math. 22, pp. 455-466 (12 pages)

Online Publication Date: March 20, 2008

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We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems under this scheme appears to be highly dependent on the choice of input type. We define an order on the various methods of description, and we show how this order acts upon 10 types of input. We also show that under this approach several natural algorithmic problems are complete in classes thought not to be equal to P.

On Ramsey Minimal Graphs

V. Rödl and M. Siggers

SIAM J. Discrete Math. 22, pp. 467-488 (22 pages)

Online Publication Date: March 20, 2008

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A graph $G$ is $r$-Ramsey-minimal with respect to a graph $H$ if every $r$-coloring of the edges of $G$ yields a monochromatic copy of $H$, but the same is not true for any proper subgraph of $G$. In this paper we show that for any integer $k \geq 3$ and $r \geq 2$, there exists a constant $c>1$ such that for large enough $n$, there exist at least $c^{n^2}$ nonisomorphic graphs on at most $n$ vertices, each of which is $r$-Ramsey-minimal with respect to the complete graph $K_k$. Furthermore, in the case $r=2$, we give an asymmetric version of the above result.

Graph-Different Permutations

János Körner, Claudia Malvenuto, and Gábor Simonyi

SIAM J. Discrete Math. 22, pp. 489-499 (11 pages) | Cited 1 time

Online Publication Date: March 20, 2008

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For a finite graph $G$ whose vertices are different natural numbers we call two infinite permutations of the natural numbers $G$-different if they have two adjacent vertices of $G$ somewhere in the same position. The maximum number of pairwise $G$-different permutations of the naturals is always finite. We study this maximum as a graph invariant and relate it to a problem of the first two authors on colliding permutations. An improvement on the lower bound for the maximum number of pairwise colliding permutations is obtained.

Realizing Degree Sequences with Graphs Having Nowhere-Zero 3-Flows

Rong Luo, Rui Xu, Wenan Zang, and Cun-Quan Zhang

SIAM J. Discrete Math. 22, pp. 500-519 (20 pages)

Online Publication Date: March 20, 2008

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The following open problem was proposed by Archdeacon: Characterize all graphical sequences $\pi$ such that some realization of $\pi$ admits a nowhere-zero 3-flow. The purpose of this paper is to resolve this problem and present a complete characterization: A graphical sequence $\pi = (d_1,d_2,\dots,d_n)$ with minimum degree at least two has a realization that admits a nowhere-zero 3-flow if and only if $\pi \neq (3^4,2)$, $(k,3^k)$, $(k^2,3^{k-1})$, where $k$ is an odd integer.

Hamilton Cycles in Random Lifts of Directed Graphs

Prasad Chebolu and Alan Frieze

SIAM J. Discrete Math. 22, pp. 520-540 (21 pages)

Online Publication Date: March 20, 2008

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An $n$-lift of a digraph $K$ is a digraph with a vertex set $V(K)\times [n]$, and for each directed edge $(i,j)\in E(K)$ there is a perfect matching between fibers $\{i\}\times [n]$ and $\{j\}\times [n]$, with edges directed from fiber $i$ to fiber $j$. If these matchings are chosen independently and uniformly at random, then we say that we have a random $n$-lift. We show that if $h$ is sufficiently large, then a random $n$-lift of the complete digraph $\DK_h$ is a Hamiltonian .

Coloring of Triangle-Free Graphs on the Double Torus

Daniel Král' and Matěj StehlÍk

SIAM J. Discrete Math. 22, pp. 541-553 (13 pages)

Online Publication Date: March 20, 2008

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We show that every triangle-free graph on the double torus is $4$-colorable. This settles a problem raised by Gimbel and Thomassen [Trans. Amer. Math. Soc., 349 (1997), pp. 4555–4564].

Power Domination in Product Graphs

Paul Dorbec, Michel Mollard, Sandi Klavžar, and Simon Špacapan

SIAM J. Discrete Math. 22, pp. 554-567 (14 pages)

Online Publication Date: March 21, 2008

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The power system monitoring problem asks for as few as possible measurement devices to be put in an electric power system. The problem has a graph theory model involving power dominating sets in graphs. The power domination number $\gamma_P(G)$ of $G$ is the minimum cardinality of a power dominating set. Dorfling and Henning [Discrete Appl. Math., 154 (2006), pp. 1023–1027] determined the power domination number of the Cartesian product of paths. In this paper the power domination number is determined for all direct products of paths except for the odd component of the direct product of two odd paths. For instance, if $n$ is even and $C$ a connected component of $P_m\times P_n$, where $m$ is odd or $m\geq n$, then $\gamma_P(C)=\left\lceil n/4 \right\rceil$. For the strong product we prove that $\gamma_P(P_n \boxtimes P_m) = \max\{\lceil n/3\rceil, \lceil (n+m-2)/4\rceil\}$, unless $3m-n-6 \equiv 4\pmod 8$. The power domination number is also determined for an arbitrary lexicographic product.

Planar Graphs of Odd-Girth at Least $9$ are Homomorphic to the Petersen Graph

Z. Dvořák, R. Škrekovski, and T. Valla

SIAM J. Discrete Math. 22, pp. 568-591 (24 pages)

Online Publication Date: March 21, 2008

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Let $G$ be a graph and let $c: V(G)\to\binom{1,\ldots,5}{2}$ be an assignment of $2$-element subsets of the set $1,\ldots,5$ to the vertices of $G$ such that for every edge $vw$, the sets $c(v)$ and $c(w)$ are disjoint. We call such an assignment a $(5,2)$-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph $G$ is the length of the shortest odd cycle in $G$ ($\infty$ if $G$ is bipartite). We prove that every planar graph of odd-girth at least $9$ is $(5,2)$-colorable, and thus it is homomorphic to the Petersen graph. Also, this implies that such graphs have a fractional chromatic number at most $5\over2$. As a special case, this result holds for planar graphs of girth at least $8$.

Hat Guessing Games

Steve Butler, Mohammad T. Hajiaghayi, Robert D. Kleinberg, and Tom Leighton

SIAM J. Discrete Math. 22, pp. 592-605 (14 pages) | Cited 1 time

Online Publication Date: March 21, 2008

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Hat problems have become a popular topic in recreational mathematics. In a typical hat problem, each of $n$ players tries to guess the color of the hat he or she is wearing by looking at the colors of the hats worn by some of the other players. In this paper we consider several variants of the problem, united by the common theme that the guessing strategies are required to be deterministic and the objective is to maximize the number of correct answers in the worst case. We also summarize what is currently known about the worst-case analysis of deterministic hat guessing problems with a finite number of players.

The Windy General Routing Polyhedron: A Global View of Many Known Arc Routing Polyhedra

Angel Corberán, Isaac Plana, and José M. Sanchis

SIAM J. Discrete Math. 22, pp. 606-628 (23 pages)

Online Publication Date: March 21, 2008

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The windy postman problem consists of finding a minimum cost traversal of all of the edges of an undirected graph with two costs associated with each edge, representing the costs of traversing it in each direction. In this paper we deal with the windy general routing problem (WGRP), in which only a subset of edges must be traversed and a subset of vertices must be visited. This is also an ${\it NP}$-hard problem that generalizes many important arc routing problems (ARPs) and has some interesting real-life applications. Here we study the description of the WGRP polyhedron, for which some general properties and some large families of facet-inducing inequalities are presented. Moreover, since the WGRP contains many well-known routing problems as special cases, this paper also provides a global view of their associated polyhedra. Finally, for the first time, some polyhedral results for several ARPs defined on mixed graphs formulated by using two variables per edge are presented.

On Complexity of the Subpattern Problem

Shlomo Ahal and Yuri Rabinovich

SIAM J. Discrete Math. 22, pp. 629-649 (21 pages)

Online Publication Date: March 28, 2008

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We study various computational aspects of the problem of determining whether, given a (fixed) permutation $\pi$ on $k$ elements and an input permutation $\sigma$ on $n > k$ elements, $\pi$ can be embedded in $\sigma$ in an order-preserving manner. Formally, the goal is to determine whether there exists a strictly increasing function $f$ from $[1,k]$ to $[1,n]$ which is order preserving, i.e., $f$ satisfies $\sigma(f(i)) > \sigma(f(j))$ whenever $\pi(i) > \pi(j)$. We call this decision problem the subpattern problem. The study falls into two parts. In the first part we develop and analyze an algorithmic paradigm for this problem. We introduce two naturally defined (related) permutation-complexity measures $C(\pi)$ and a somewhat finer $C^{\bf T}(\pi)$, and, we show that our algorithms run in time $O(n^{1 + C(\pi)})$ and $O(n^{2 \cdot C^{\bf T}(\pi)})$, respectively; i.e., the hardness of the problem crucially depends on the structure of $\pi$, as measured by $C(\pi)$ or by $C^{\bf T}(\pi)$. In the second part of the paper we study the above complexity measures. In particular, we show that in the general case, $C(\pi) \leq 0.47k + o(k)$. Thus, the time complexity of the subpattern problem is at most $O(n^{0.47k + o(k)})$, improving over the trivial $O(n^k)$. Unfortunately, it turns out that for most permutations $C^{\bf T}(\pi) = \Omega(k)$, and thus, in general, the upper bound on the running time cannot be significantly improved using this approach. Yet, for many natural classes of permutations the complexity of $C(\pi)$ is sublinear in $k$. To demonstrate this, we study two interesting classes of “linear” permutations and show that their complexity is $C(\pi) = O(\sqrt{k})$. In addition, we study some structural properties of the complexity measures, show that $C^{\bf T}(\pi) \leq C(\pi) \leq O(\log k) \cdot C^{\bf T}(\pi)$, and relate $C(\pi)$ and $C^{\bf T}(\pi)$ to the pathwidth and the treewidth of a certain graph $G_\pi$ defined by the permutation $\pi$.

Cubic Monomial Bent Functions: A Subclass of $\mathcal{M}$

Pascale Charpin and Gohar M. Kyureghyan

SIAM J. Discrete Math. 22, pp. 650-665 (16 pages)

Online Publication Date: March 28, 2008

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Based on a computer search, Anne Canteaut conjectured that the exponent $2^{2r}+2^r+1$ in ${\bf F}_{2^{6r}}$ and the exponent $(2^{r}+1)^2$ in ${\bf F}_{2^{4r}}$ yield bent monomial functions. These conjectures are proved in [A. Canteaut, P. Charpin, and G. Kyureghyan, A new class of monomial bent functions, in Proceedings of the 2006 IEEE International Symposium on Information Theory, (ISIT 06 Seattle), IEEE Press, Piscataway, NJ, 2006, pp. 903–906] and [N. G. Leander, IEEE Trans. Inform. Theory, 52 (2006), pp. 738–743]. Both exponents are of binary weight $3$ and define functions from the Maiorana–McFarland class $\mathcal{M}$ of bent functions to the subfield. In this paper we show that these are the only such exponents. Our proof is based on the classification of the permutation binomials $X^{2^k+2}+ \nu X$ of finite fields of even characteristics. We also extend the result of Leander, determining all bent monomial functions with the exponent $(2^{r}+1)^2$.

Rank-Width and Well-Quasi-Ordering

Sang-il Oum

SIAM J. Discrete Math. 22, pp. 666-682 (17 pages)

Online Publication Date: March 28, 2008

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Robertson and Seymour [J. Combin. Theory Ser. B, 48 (1990), pp. 227–254] proved that graphs of bounded tree-width are well-quasi-ordered by the graph minor relation. By extending their arguments, Geelen, Gerards, and Whittle [J. Combin. Theory Ser. B, 84 (2002), pp. 270–290] proved that binary matroids of bounded branch-width are well-quasi-ordered by the matroid minor relation. We prove another theorem of this kind in terms of rank-width and vertex-minors. For a graph $G=(V,E)$ and a vertex $v$ of $G$, a local complementation at $v$ is an operation that replaces the subgraph induced by the neighbors of $v$ with its complement graph. A graph $H$ is called a vertex-minor of $G$ if $H$ can be obtained from $G$ by applying a sequence of vertex deletions and local complementations. Rank-width was defined by Oum and Seymour [J. Combin. Theory Ser. B, 96 (2006), pp. 514–528] to investigate clique-width; they showed that graphs have bounded rank-width if and only if they have bounded clique-width. We prove that graphs of bounded rank-width are well-quasi-ordered by the vertex-minor relation; in other words, for every infinite sequence $G_1,G_2,\ldots$ of graphs of rank-width (or clique-width) at most $k$, there exist $i<j$ such that $G_i$ is isomorphic to a vertex-minor of $G_j$. This implies that there is a finite list of graphs such that a graph has rank-width at most $k$ if and only if it contains no one in the list as a vertex-minor. The proof uses the notion of isotropic systems defined by Bouchet.

Coloring an Orthogonality Graph

C. D. Godsil and M. W. Newman

SIAM J. Discrete Math. 22, pp. 683-692 (10 pages)

Online Publication Date: March 28, 2008

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We deal with a graph coloring problem that arises in quantum information theory. Alice and Bob are each given a $\pm1$-vector of length $2^k$ and are to respond with $k$ bits. Their responses must be equal if they are given equal inputs, and distinct if they are given orthogonal inputs; however, they are not allowed to communicate any information about their inputs. They can always succeed using quantum entanglement, but their ability to succeed using only classical physics is equivalent to a graph coloring problem. We resolve the graph coloring problem, thus determining that they can succeed without entanglement exactly when $k\leq3$.

Finding a Maximum Independent Set in a Sparse Random Graph

Uriel Feige and Eran Ofek

SIAM J. Discrete Math. 22, pp. 693-718 (26 pages)

Online Publication Date: March 28, 2008

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We consider the problem of finding a maximum independent set in a random graph. The random graph $G$, which contains $n$ vertices, is modeled as follows. Every edge is included independently with probability $\frac{d}{n}$, where $d$ is some sufficiently large constant. Thereafter, for some constant $\alpha$, a subset $I$ of $\alpha n$ vertices is chosen at random, and all edges within this subset are removed. In this model, the planted independent set $I$ is a good approximation for the maximum independent set $I_{max}$, but both $I \setminus I_{max}$ and $I_{max} \setminus I$ are likely to be nonempty. We present a polynomial time algorithm that with high probability (over the random choice of random graph $G$ and without being given the planted independent set $I$) finds the maximum independent set in $G$ when $\alpha \geq \sqrt{c_0 /d}$, where $c_0$ is some sufficiently large constant independent of $d$.

A Census of Small Latin Hypercubes

Brendan D. McKay and Ian M. Wanless

SIAM J. Discrete Math. 22, pp. 719-736 (18 pages) | Cited 1 time

Online Publication Date: April 18, 2008

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We count all latin cubes of order $n\le6$ and latin hypercubes of order $n\le5$ and dimension $d\le5$. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of $n$ and $d$ we classify all $d$-ary quasigroups of order $n$ into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the $d$-ary loops). We also give an exact formula for the number of (isomorphism classes of) $d$-ary quasigroups of order 3 for every $d$. Then we give a number of constructions for $d$-ary quasigroups with a specific number of identity elements. In the process, we prove that no $3$-ary loop of order $n$ can have exactly $n-1$ identity elements (but no such result holds in dimensions other than 3). Finally, we give some new examples of latin cuboids which cannot be extended to latin cubes.

The Nine Morse Generic Tetrahedra

D. Siersma and M. van Manen

SIAM J. Discrete Math. 22, pp. 737-746 (10 pages)

Online Publication Date: April 18, 2008

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There are two types of shapes for a generic triangle—acute and obtuse. These shapes are also distinguished by the (topological) Morse theory of the minimal distance function to the vertices. We can use the same method for a tetrahedron, and we show in this paper that there exist nine generic shapes. These can be described by a Morse poset or by a Gabriel graph. We also report on some computer experiments and compare our classification to another criterion used in computational geometry.

Walkers on the Cycle and the Grid

J. Díaz, X. Pérez, M. J. Serna, and N. C. Wormald

SIAM J. Discrete Math. 22, pp. 747-775 (29 pages)

Online Publication Date: April 18, 2008

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We present a model of the establishment and maintenance of communication between mobile agents. We assume that the agents move through a fixed environment modeled by a motion graph and are able to communicate if they are within distance at most $d$ of each other. As the agents move randomly, we analyze the evolution in time of the connectivity between a set of $w$ agents, asymptotically for a large number $N$ of vertices, when $w$ also grows large. The particular topologies of the environment we study here are the cycle and the toroidal grid.

Combinatorial Properties of a Rooted Graph Polynomial

David Eisenstat, Gary Gordon, and Amanda Redlich

SIAM J. Discrete Math. 22, pp. 776-785 (10 pages)

Online Publication Date: April 18, 2008

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For a rooted graph $G$, let $EV(G;p)$ be the expected number of vertices reachable from the root when each edge has an independent probability $p$ of operating successfully. We examine combinatorial properties of this polynomial, proving that $G$ is $k$-edge connected if and only if $EV'(G;1)=\cdots=EV^{k-1}(G;1)=0$. We find bounds on the first and second derivatives of $EV(G;p)$; applications yield characterizations of rooted paths and cycles in terms of the polynomial. We prove reconstruction results for rooted trees and a negative result concerning reconstruction of more complicated rooted graphs. We also prove that the norm of the largest root of $EV(G;p)$ in $\mathbb{Q}[i]$ gives a sharp lower bound on the number of vertices of $G$.

Testing Triangle-Freeness in General Graphs

Noga Alon, Tali Kaufman, Michael Krivelevich, and Dana Ron

SIAM J. Discrete Math. 22, pp. 786-819 (34 pages) | Cited 1 time

Online Publication Date: April 18, 2008

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In this paper we consider the problem of testing whether a graph is triangle-free and, more generally, whether it is $H$-free, for a fixed subgraph $H$. The algorithm should accept graphs that are triangle-free and reject graphs that are far from being triangle-free in the sense that a constant fraction of the edges should be removed in order to obtain a triangle-free graph. The algorithm is allowed a small probability of error. This problem has been studied quite extensively in the past, but the focus was on dense graphs, that is, when $d = \Theta(n)$, where $d$ is the average degree in the graph and $n$ is the number of vertices. Here we study the complexity of the problem in general graphs, that is, for varying $d$. In this model a testing algorithm is allowed to ask neighbor queries (i.e., “What is the $i$th neighbor of vertex $v$?”), vertex-pair queries (i.e., “Is there an edge between vertices $v$ and $u$?”), and degree queries (i.e., “What is the degree of vertex $v$?”). Our main finding is a lower bound of $\Omega(n^{1/3})$ on the necessary number of queries that holds for every $d < n^{1-\nu(n)}$, where $\nu(n) = o(1)$. Since when $d = \Theta(n)$ the number of queries sufficient for testing has been known to be independent of $n$, we observe an abrupt, threshold-like behavior of the complexity of testing around $n$. This lower bound holds for testing $H$-freeness of every nonbipartite subgraph $H$. Additionally, we provide sublinear upper bounds for testing triangle-freeness that are at most quadratic in the stated lower bounds, and we describe a transformation from certain one-sided error lower bounds for testing subgraph-freeness to two-sided error lower bounds. Finally, in the course of our analysis we show that dense random Cayley graphs behave like quasi-random graphs in the sense that relatively large subsets of vertices have the “correct” edge density. The result for subsets of this size cannot be obtained from the known spectral techniques that only supply such estimates for much larger subsets.

A Note On Reed's Conjecture

Landon Rabern

SIAM J. Discrete Math. 22, pp. 820-827 (8 pages)

Online Publication Date: April 25, 2008

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In [J. Graph Theory, 27 (1998), pp. 177–212], Reed conjectures that every graph satisfies $\chi \leq \lceil \frac{\omega + \Delta + 1}{2} \rceil$. We prove that this holds for graphs with disconnected complement. Combining this fact with a result of Molloy proves the conjecture for graphs satisfying $\chi > \lceil\frac{n}{2}\rceil$. Generalizing this we prove that the conjecture holds for graphs satisfying $\chi > \frac{n + 3 - \alpha}{2}$. It follows that the conjecture holds for graphs satisfying $\Delta \geq n + 2 - (\alpha + \sqrt{n + 5 - \alpha})$. In the final section, we show that if $G$ is an even order counterexample to Reed's conjecture, then $\overline{G}$ has a $1$-factor.

The Minimum Number of Distinct Areas of Triangles Determined by a Set of $n$ Points in the Plane

Rom Pinchasi

SIAM J. Discrete Math. 22, pp. 828-831 (4 pages)

Online Publication Date: April 25, 2008

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We prove a conjecture of Erdős, Purdy, and Straus on the number of distinct areas of triangles determined by a set of $n$ points in the plane. We show that if $P$ is a set of $n$ points in the plane, not all on one line, then $P$ determines at least $\lfloor\frac{n-1}{2}\rfloor$ triangles with pairwise distinct areas. Moreover, one can find such $\lfloor\frac{n-1}{2}\rfloor$ triangles all sharing a common edge.

On the Complexity of Ordered Colorings

Arvind Gupta, Jan van den Heuvel, Ján Maňuch, Ladislav Stacho, and Xiaohong Zhao

SIAM J. Discrete Math. 22, pp. 832-847 (16 pages)

Online Publication Date: April 25, 2008

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We introduce two variants of proper colorings with imposed partial ordering on the set of colors. One variant shows very close connections to some fundamental problems in graph theory, e.g., directed graph homomorphism and list colorings. We study the border between tractability and intractability for both variants.
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