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2007

Volume 21, Issue 1, pp. 1-272


A New Min‐Cut Max‐Flow Ratio for Multicommodity Flows

Oktay Günlük

SIAM J. Discrete Math. 21, pp. 1-15 (15 pages)

Online Publication Date: January 08, 2007

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In this paper we present a new bound on the min‐cut max‐flow ratio for multicommodity flow problems with specified demands. For multicommodity flows, this is a generalization of the well‐known relationship between the capacity of a minimum cut and the value of the maximum flow of a single commodity flow problem. For multicommodity flows, capacity of a cut is scaled by the demand that has to cross the cut to obtain the numerator of this ratio. In the denominator, the maximum concurrent flow value is used. Currently, the best known bound for this ratio is proportional to $\log(k)$, where $k$ is the number of origin‐destination pairs with positive demand. Our new bound is proportional to $\log(k^*)$, where $k^*$ is the cardinality of the minimum cardinality vertex cover of the demand graph. To obtain this bound, we start with a so‐called aggregated commodity formulation of the maximum concurrent flow problem with $k^*$ commodities. We also show a similar bound for the maximum multicommodity flow problem. The new bound is proportional to $\min\{\log(k^*), k^{**}\}$, where $k^{**}$ denotes the size of the of the demand graph.

A Simple Gray Code to List All Minimal Signed Binary Representations

J. Sawada

SIAM J. Discrete Math. 21, pp. 16-25 (10 pages)

Online Publication Date: January 08, 2007

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A signed binary representation (SBR) of an integer $N$ is a string $a_b\cdots a_2a_1a_0$ over the alphabet $\{-1,0,1\}$ such that $N = \sum_{i=0}^b a_i2^i$. An SBR of an integer $N$ is said to be minimal if the number of nonzero digits is minimum. In this paper, we describe a simple 3–close Gray code for listing all minimal SBRs of an integer $N$. The algorithm is implemented to run in constant amortized time. In addition, we identify the values for $N$ that have the maximum number of minimal SBRs given the length of the binary representation of $N$.

Crossing Graphs as Joins of Graphs and Cartesian Products of Median Graphs

Boštjan Brešar and Sandi Klavžar

SIAM J. Discrete Math. 21, pp. 26-32 (7 pages)

Online Publication Date: January 08, 2007

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For a partial cube $G$ its crossing graph $G^\#$ is the graph whose vertices are the ϴ‐classes of $G$, two classes being adjacent if they cross on some cycle in $G$. The following problem posed in [S. Klavžar and H. M. Mulder, SIAM J. Discrete Math., 15 (2002), pp. 235–251, Problem 7.1] is considered: What can be said about the partial cube $G$ if $G^\#$ is the join $A\oplus B$ of graphs $A$ and $B$ with at least one edge? It is proved that for arbitrary graphs $A$ and $B$, where at least one of them contains an edge, there exists a Cartesian prime partial cube $G$ such that $G^\# = A\oplus B$. On the other hand, if $G$ is a median graph, then $G^\# = A\oplus B$ if and only if $G=H\,\square\, K$, where $H^\# = A$ and $K^\# = B$. Along the way some new facts about partial cubes are obtained; for instance, a bipartite graph of radius 2 is a partial cube if and only if it is $K_{2,3}$‐free.

Correlation of Graph‐Theoretical Indices

Stephan G. Wagner

SIAM J. Discrete Math. 21, pp. 33-46 (14 pages) | Cited 4 times

Online Publication Date: January 12, 2007

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The correlation of graph characteristics, such as the number of independent vertex or edge subsets, the number of connected subsets, or the sum of distances, which also play a role in combinatorial chemistry, is studied by a generating function approach and asymptotic analysis. It is shown how an asymptotic formula for the correlation coefficient can be obtained when simply generated families of trees are investigated. For rooted ordered trees, the calculations are done explicitly. Further feasible correlation measures are discussed.

The Spectrum of the Corona of Two Graphs

S. Barik, S. Pati, and B. K. Sarma

SIAM J. Discrete Math. 21, pp. 47-56 (10 pages) | Cited 5 times

Online Publication Date: January 12, 2007

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We consider only simple graphs. Given two graphs $G$ with vertices $1,\ldots,n$ and $H$, the corona $G\circ H$ is defined as the graph obtained by taking $n$ copies of $H$ and for each $i$ inserting edges between the $i$th vertex of $G$ and each vertex of the $i$th copy of $H$. For a connected graph $G$ and any $r$‐regular graph $H$ we provide complete information about the spectrum of $G\circ H$ using the spectrum of $G$ and spectrum of $H$. Complete information about the Laplacian spectrum of $G\circ H$ is also provided even when $H$ is not regular. A graph $G$ is said to have the property (R) if $\frac{1}{\lambda}$ is an eigenvalue of $G$ whenever $\lambda$ is an eigenvalue of $G$. Further, if $\lambda$ and $\frac{1}{\lambda}$ have the same multiplicity, for each eigenvalue $\lambda$, then it is said to have the property (SR). We characterize all trees with property (SR) and show that such a tree is the corona product of some tree and an isolated vertex. We supply a family of bipartite graphs with property (R). As an application we construct infinitely many pairs of nonisomorphic graphs with the same spectrum and the same Laplacian spectrum. We prove some results about the eigenvector related to the second smallest eigenvalue of the Laplacian matrix of $G\circ H$ and give an application.

Sharp Threshold for Hamiltonicity of Random Geometric Graphs

Josep Díaz, Dieter Mitsche, and Xavier Pérez

SIAM J. Discrete Math. 21, pp. 57-65 (9 pages)

Online Publication Date: January 26, 2007

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We show for an arbitrary $\ell_p$ norm that the property that a random geometric graph $\mathcal G(n,r)$ contains a Hamiltonian cycle exhibits a sharp threshold at $r=r(n)=\sqrt{\frac{\log n}{\alpha_p n}}$, where $\alpha_p$ is the area of the unit disk in the $\ell_p$ norm. The proof is constructive and yields a linear time algorithm for finding a Hamiltonian cycle of $\mathcal{G}(n,r)$ asymptotically almost surely, provided $r=r(n)\ge\sqrt{\frac{\logn}{(\alpha_p -\epsilon)n}}$ for some fixed $\epsilon>0$.

Turán’s Theorem in the Hypercube

Noga Alon, Anja Krech, and Tibor Szabó

SIAM J. Discrete Math. 21, pp. 66-72 (7 pages)

Online Publication Date: February 05, 2007

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We are motivated by the analogue of Turán’s theorem in the hypercube $Q_n$: How many edges can a $Q_d$‐free subgraph of $Q_n$ have? We study this question through its Ramsey‐type variant and obtain asymptotic results. We show that for every odd $d$ it is possible to color the edges of $Q_n$ with $\frac{(d+1)^2}{4}$ colors such that each subcube $Q_d$ is polychromatic, that is, contains an edge of each color. The number of colors is tight up to a constant factor, as it turns out that a similar coloring with ${d+1\choose 2} +1$ colors is not possible. The corresponding question for vertices is also considered. It is not possible to color the vertices of $Q_n$ with $d+2$ colors such that any $Q_d$ is polychromatic, but there is a simple $d+1$ coloring with this property. A relationship to anti‐Ramsey colorings is also discussed. We discover much less about the Turán‐type question which motivated our investigations. Numerous problems and conjectures are raised.

Every Monotone $3$‐Graph Property is Testable

Christian Avart, Vojtěch Rödl, and Mathias Schacht

SIAM J. Discrete Math. 21, pp. 73-92 (20 pages) | Cited 1 time

Online Publication Date: February 05, 2007

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Recently Alon and Shapira [Every monotone graph property is testable, New York, Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, ACM Press, 2005, pp. 128–137] have established that every monotone graph property is testable. They raised the question whether their results can be extended to hypergraphs. The aim of this paper is to address this problem. Based on the recent regularity lemma of Rödl and Schacht [Regular partitions of hypergraphs, Combin. Probab. Comput., to appear], we prove that any monotone property of 3‐uniform hypergraphs is testable answering in part the question of Alon and Shapira. Our approach is similar to the one developed by Alon and Shapira for graphs. We believe that based on the general version of the hypergraph regularity lemma the proof presented in this article extends to $k$‐uniform hypergraphs.

A Generalization of Kotzig’s Theorem and Its Application

Richard Cole, Łukasz Kowalik, and Riste Škrekovski

SIAM J. Discrete Math. 21, pp. 93-106 (14 pages)

Online Publication Date: February 09, 2007

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An edge of a graph is light when the sum of the degrees of its end‐vertices is at most 13. The well‐known Kotzig theorem states that every 3‐connected planar graph contains a light edge. Later, Borodin [J. Reine Angew. Math., 394 (1989), pp. 180–185] extended this result to the class of planar graphs of minimum degree at least 3. We deal with generalizations of these results for planar graphs of minimum degree 2. Borodin, Kostochka, and Woodall [J. Combin. Theory Ser. B, 71 (1997), pp. 184–204] showed that each such graph contains a light edge or a member of two infinite sets of configurations, called 2‐alternating cycles and 3‐alternators. This implies that planar graphs with maximum degree $\Delta \geq 12$ are $\Delta$‐edge‐choosable. We prove a similar result with 2‐alternating cycles and 3‐alternators replaced by five fixed bounded‐sized configurations called crowns. This gives another proof of $\Delta$‐edge‐choosability of planar graphs with $\Delta \geq 12$. However, we show efficient choosability; i.e., we describe a linear‐time algorithm for $\max\{\Delta,12\}$‐edge‐list‐coloring planar graphs. This extends the result of Chrobak and Yung [J. Algorithms, 10 (1989), pp. 35–51].

Operations on M‐Convex Functions on Jump Systems

Yusuke Kobayashi, Kazuo Murota, and Ken’ichiro Tanaka

SIAM J. Discrete Math. 21, pp. 107-129 (23 pages)

Online Publication Date: February 09, 2007

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A jump system is a set of integer points with an exchange property, which is a generalization of a matroid, a delta‐matroid, and a base polyhedron of an integral polymatroid (or a submodular system). Recently, the concept of M‐convex functions on constant‐parity jump systems was introduced by Murota as a class of discrete convex functions that admit a local criterion for global minimality. M‐convex functions on constant‐parity jump systems generalize valuated matroids, valuated delta‐matroids, and M‐convex functions on base polyhedra. This paper reveals that the class of M‐convex functions on constant‐parity jump systems is closed under a number of natural operations such as splitting, aggregation, convolution, composition, and transformation by networks. The present results generalize hitherto‐known similar constructions for matroids, delta‐matroids, valuated matroids, valuated delta‐matroids, and M‐convex functions on base polyhedra.

A Natural Family of Flag Matroids

Anna de Mier

SIAM J. Discrete Math. 21, pp. 130-140 (11 pages) | Cited 1 time

Online Publication Date: February 15, 2007

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A flag matroid can be viewed as a chain of matroids linked by quotients. Flag matroids, of which relatively few interesting families have previously been known, are a particular class of Coxeter matroids. In this paper we give a family of flag matroids arising from an enumeration problem that is a generalization of the tennis ball problem. These flag matroids can also be defined in terms of lattice paths, and they provide a generalization of the lattice path matroids of [J. Bonin, A. de Mier, and M. Noy, J. Combin. Theory Ser. A, 104 (2003), pp. 63–94].

Algorithms for Fault‐Tolerant Routing in Circuit‐Switched Networks

Amitabha Bagchi, Amitabh Chaudhary, Christian Scheideler, and Petr Kolman

SIAM J. Discrete Math. 21, pp. 141-157 (17 pages)

Online Publication Date: February 15, 2007

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In this paper we consider the $k$ edge‐disjoint paths problem ($k$‐EDP), a generalization of the well‐known edge‐disjoint paths problem. Given a graph $G=(V,E)$ and a set of terminal pairs (or requests) $T$, the problem is to find a maximum subset of the pairs in $T$ for which it is possible to select paths such that each pair is connected by $k$ edge‐disjoint paths and the paths for different pairs are mutually disjoint. To the best of our knowledge, no nontrivial result is known for this problem for $k>1$. To measure the performance of our algorithms we use the recently introduced flow number $F$ of a graph. This parameter is known to fulfill $F=O(\Delta \alpha^{-1} \logn)$, where $\Delta$ is the maximum degree, $\alpha$ is the edge expansion of $G$, and $n$ is the number of vertices in $G$. We show that a simple greedy online algorithm achieves a competitive ratio of $O(k^3 F)$ which naturally extends the best known bound of $O(F)$ for $k=1$ to higher $k$. To achieve this competitive ratio, we introduce a new method of converting a system of $k$ disjoint paths into a system of $k$ length‐bounded disjoint paths. We also show that any deterministic online algorithm has a competitive ratio of $\Omega(k F)$. In addition, we study the $k$ disjoint flows problem ($k$‐DFP), which is a generalization of the previously studied unsplittable flow problem. The difference between the $k$‐DFP and the $k$‐EDP is that now we consider a graph with edge capacities and our requests are allowed to have arbitrary demands $d_i$. The aim is to find a subset of requests of maximum total demand for which it is possible to select flow paths such that all the capacity constraints are maintained and each selected request with demand $d_i$ is connected by $k$ disjoint paths, each of flow value $d_i/k$. The $k$‐EDP and $k$‐DFP problems have important applications in fault‐tolerant (virtual) circuit switching, which plays a key role in optical networks.

The Complexity of Combinatorial Optimization Problems on $d$‐Dimensional Boxes

Miroslav Chlebík and Janka Chlebíková

SIAM J. Discrete Math. 21, pp. 158-169 (12 pages)

Online Publication Date: February 26, 2007

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The MAXIMUM INDEPENDENT SET problem in $d$‐box graphs, i.e., in intersection graphs of axis‐parallel rectangles in $\mathbb{R}^d$, is known to be NP‐hard for any fixed $d\geq 2$. A challenging open problem is that of how closely the solution can be approximated by a polynomial time algorithm. For the restricted case of $d$‐boxes with bounded aspect ratio a PTAS exists [T. Erlebach, K. Jansen, and E. Seidel, SIAM J. Comput., 34 (2005), pp. 1302–1323]. In the general case no polynomial time algorithm with approximation ratio $o(\log^{d-1} n)$ for a set of $n$ $d$‐boxes is known. In this paper we prove APX‐hardness of the MAXIMUM INDEPENDENT SET problem in $d$‐box graphs for any fixed $d\geq 3$. We give an explicit lower bound $\frac{245}{244}$ on efficient approximability for this problem unless $\PP=\text{\rm NP}$. Additionally, we provide a generic method how to prove APX‐hardness for other graph optimization problems in $d$‐box graphs for any fixed $d\geq 3$.

Vertex‐Magic Total Labelings of Regular Graphs

Ian D. Gray

SIAM J. Discrete Math. 21, pp. 170-177 (8 pages) | Cited 2 times

Online Publication Date: March 02, 2007

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In this paper it is shown that if a graph $G$ possesses a spanning subgraph $H$ with a strong vertex magic total labeling (VMTL) and $G-E(H)$ is even‐regular, then $G$ also has a strong VMTL. Among other things, this is used to conclude that all Hamiltonian regular graphs of odd order possess strong VMTLs. A relationship is then demonstrated between regular graphs of even degree and sparse magic squares. We next consider cubic graphs of order $2n$ consisting of two 2‐factors of order $n$, connected by a 1‐factor (quasi‐prisms). Based on McQuillan’s construction of VMTLs of such 3‐regular graphs, VMTLs are derived for similar regular graphs of any odd degree. Finally, a construction is given for VMTLs of quartic graphs of order $4n+2$ consisting of two cycles of odd order $n$ connected by a 2‐factor (simple quasi‐anti‐prisms), and based on this construction VMTLs are derived for similar regular graphs of any even degree.

Two New Bounds for the Random‐Edge Simplex‐Algorithm

Bernd Gärtner and Volker Kaibel

SIAM J. Discrete Math. 21, pp. 178-190 (13 pages)

Online Publication Date: March 15, 2007

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We prove that the RANDOM‐EDGE simplex‐algorithm requires an expected number of at most $13n/\sqrt{d}$ pivot steps on any simple $d$‐polytope with $n$ vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial $d$‐cubes the trivial upper bound of $2^d$ on the performance of RANDOM‐EDGE can asymptotically be improved by the factor $1/d^{(1-\varepsilon)\log d}$ for every $\varepsilon>0$.

Quadratically Many Colorful Simplices

Imre Bárány and Jiří Matoušek

SIAM J. Discrete Math. 21, pp. 191-198 (8 pages) | Cited 3 times

Online Publication Date: March 15, 2007

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The colorful Carathéodory theorem asserts that if $X_1,X_2,\ldots,X_{d+1}$ are sets in ${\bf R}^d$, each containing the origin 0 in its convex hull, then there exists a set $S \subseteq X_1 \cup \cdots \cup X_{d+1}$ with $|S \cap X_i| = 1$ for all $i=1,2,\ldots,d+1$ and $0 \in conv(S)$ (we call $conv(S)$ a colorful covering simplex). Deza et al. [Discrete Comput. Geom., 35 (2006), pp. 597–615] proved that if the $X_i$ are in general position with respect to 0 (consequently, each $X_i$ has at least $d+1$ points), then there are at least $2d$ colorful covering simplices, and they constructed an example with no more than $d^2+1$ such simplices. Under the same assumption, we show that there are at least $\frac{1}{5}d(d+1)$ colorful covering simplices, thus determining the order of magnitude. A similar result was proved independently by Stephen and Thomas [http://www.arxiv.org/abs/math.CO/0512400 (2005)]. We also obtain a lower bound of $3d$ for $d \geq 3$, which is better for small $d$ and, in particular, together with a parity argument it settles the case $d=3$, where the minimum possible number of colorful covering simplices is 10.

On Nearly Orthogonal Lattice Bases and Random Lattices

Ramesh Neelamani, Sanjeeb Dash, and Richard G. Baraniuk

SIAM J. Discrete Math. 21, pp. 199-219 (21 pages)

Online Publication Date: March 15, 2007

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We study lattice bases where the angle between any basis vector and the linear subspace spanned by the other basis vectors is at least $\frac{\pi}{3}$ radians; we denote such bases as “nearly orthogonal.” We show that a nearly orthogonal lattice basis always contains a shortest lattice vector. Moreover, we prove that if the basis vector lengths are “nearly equal,” then the basis is the unique nearly orthogonal lattice basis up to multiplication of basis vectors by $\pm 1$. We also study random lattices generated by the columns of random matrices with $n$ rows and $m \leq n$ columns. We show that if $m \leq c\,n$, with $c \approx 0.071$, then the random matrix forms a nearly orthogonal basis for the random lattice with high probability for large $n$ and almost surely as $n$ tends to infinity. Consequently, the columns of such a random matrix contain the shortest vector in the random lattice. Finally, we discuss an interesting JPEG image compression application where nearly orthogonal lattice bases play an important role.

A Primal Barvinok Algorithm Based on Irrational Decompositions

Matthias Köppe

SIAM J. Discrete Math. 21, pp. 220-236 (17 pages) | Cited 5 times

Online Publication Date: March 22, 2007

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We introduce variants of Barvinok’s algorithm for counting lattice points in polyhedra. The new algorithms are based on irrational signed decomposition in the primal space and the construction of rational generating functions for cones with low index. We give computational results that show that the new algorithms are faster than the existing algorithms by a large factor.

Adjacent Vertex Distinguishing Edge‐Colorings

P. N. Balister, E. Győri, J. Lehel, and R. H. Schelp

SIAM J. Discrete Math. 21, pp. 237-250 (14 pages) | Cited 8 times

Online Publication Date: April 06, 2007

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An adjacent vertex distinguishing edge‐coloring of a simple graph $G$ is a proper edge‐coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors $\chi^\prime_a(G)$ required to give $G$ an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove $\chi^\prime_a(G)\le5$ for such graphs with maximum degree $\Delta(G)=3$ and prove $\chi^\prime_a(G)\le\Delta(G)+2$ for bipartite graphs. These bounds are tight. For $k$‐chromatic graphs $G$ without isolated edges we prove a weaker result of the form $\chi^\prime_a(G)=\Delta(G)+O(\log k)$.

On the Minimum Order of Extremal Graphs to have a Prescribed Girth

C. Balbuena and P. García–Vázquez

SIAM J. Discrete Math. 21, pp. 251-257 (7 pages) | Cited 2 times

Online Publication Date: April 06, 2007

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We show that any $n$‐vertex extremal graph $G$ without cycles of length at most $k$ has girth exactly $k+1$ if $k\ge 6$ and $n>(2(k-2)^{k-2}+k-5)/(k-3)$. This result provides an improvement of the asymptotical known result by Lazebnik and Wang [J. Graph Theory, 26 (1997), pp. 147–153] who proved that the girth is exactly $k+1$ if $k\ge 12$ and $n\ge 2^{a^2+a+1}k^a$, where $a=k-3-\lfloor(k-2)/4\rfloor$. Moreover, we prove that the girth of $G$ is at most $k+2$ if $n>(2(t-2)^{k-2}+t-5)/(t-3)$, where $t=\lceil (k+1)/2\rceil\ge 4$. In general, for $k\ge 5$ we show that the girth of $G$ is at most $2k-4$ if $n\ge 2k-2$.

Precoloring Extension for 2‐connected Graphs

Margit Voigt

SIAM J. Discrete Math. 21, pp. 258-263 (6 pages) | Cited 1 time

Online Publication Date: April 06, 2007

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Let $G=G(V,E)$ be a simple graph, L a list assignment with $|L(v)|=\Delta(G)$ for all $v\in V$, and $W \subseteq V$ an independent subset of the vertex set. Define $d(W):= {\rm min} \{ d(v,w) | v,w \in W \}$ to be the minimum distance between two vertices of $W$. In this paper it is shown that if $G$ is 2‐connected with $\Delta(G) \geq 4$ and $G$ is not the complete graph $K_{\Delta(G)+1}$, then every precoloring of $W$ is extendable to a proper list coloring of $G$ provided that $d(W)\geq 4$. An example shows that the bound is sharp. This extends a result of Axenovich [Electron. J. Combin., 10 (2003), note 1] and Albertson, Kostochka, and West [SIAM J. Discrete Math., 18 (2004), pp. 542–553], who proved that $d(W)\geq 8$ guarantees such an extension for all $G$ with $\Delta(G)\geq 3$ not containing $K_{\Delta(G)+1}$.

Graphs Having Small Number of Sizes on Induced k‐Subgraphs

Maria Axenovich and József Balogh

SIAM J. Discrete Math. 21, pp. 264-272 (9 pages) | Cited 2 times

Online Publication Date: April 10, 2007

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Let $\ell$ be any positive integer, let $n$ be a sufficiently large number, and let $G$ be a graph on $n$ vertices. Define, for any $k$, $\nu_k(G)= | \{ |E(H)| : H$ is an induced subgraph of $G$ on $k$ vertices$\} |$. We show that if there exists a $k$, $2\ell \leq k \leq n-2\ell$, such that $\nu_k(G) \le \ell$, then $G$ has a complete or an empty subgraph on at least $n-\ell+1$ vertices and a homogeneous set of size at least $n-2\ell+2$. These results are sharp.
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