Showing papers in "Combinatorica in 2014"

Shorter tours by nicer ears: 7/5-Approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs

Abstract: We prove new results for approximating the graph-TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graph-TSP itself, we improve the approximation ratio to 7=5. For a generalization, the minimum T-tour problem, we obtain the first nontrivial approximation algorithm, with ratio 3=2. This contains the s-t-path graph-TSP as a special case. Our approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4=3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.

A minimum degree condition forcing complete graph immersion

Abstract: An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P uv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths P uv are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd 3/2, where c>0 is an absolute constant. For small values of t, 1≤t≤7, every simple graph of minimum degree at least t−1 contains an immersion of K t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

Connectivity and tree structure in finite graphs

Abstract: Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditionsunder which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph. As an application, we show that the k-blocks -- the maximal vertex sets that cannot be separated by at most k vertices -- of a graph G live in distinct parts of a suitable treedecomposition of G of adhesion at most k, whose decomposition tree is invariant under the automorphisms of G. This extends recent work of Dunwoody and Kron and, like theirs, generalizes a similar theorem of Tutte for k=2. Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all k simultaneously, all the k-blocks of a finite graph.

Exact solution of the hypergraph Turán problem for k-uniform linear paths

Abstract: A k-uniform linear path of length l, denoted by ℙ l (k) , is a family of k-sets {F 1,...,F l such that |F i ∩ F i+1|=1 for each i and F i ∩ F bj = $ ot 0$ whenever |i−j|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turan number of H, denoted by ex k (n, H), is the maximum number of edges in a k-uniform hypergraph $\mathcal{F}$ on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex k (n, P l (k) exactly for all fixed l ≥1, k≥4, and sufficiently large n. We show that $ex_k (n,\mathbb{P}_{2t + 1}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} )$ . The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that $ex(n,\mathbb{P}_{2t + 2}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} ) + (_{k - 2}^{n - t - 2} )$ , and describe the unique extremal family. Stability results on these bounds and some related results are also established.

Ore's conjecture for k=4 and Grötzsch's Theorem

Abstract: A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k−1)-colorable. Let f k (n) denote the minimum number of edges in an n-vertex k-critical graph. In a very recent paper, we gave a lower bound, f k (n)≥(k, n), that is sharp for every n≡1 (mod k−1). It is also sharp for k=4 and every n≥6. In this note, we present a simple proof of the bound for k=4. It implies the case k=4 of two conjectures: Gallai in 1963 conjectured that if n≡1 (mod k−1) then $f_k (n)\tfrac{{(k + 1)(k - 2)n - k(k - 3)}} {{2(k - 1)}}$ , and Ore in 1967 conjectured that for every k≥4 and $n \geqslant k + 2,f_k (n + k - 1) = f(n) + \tfrac{{k - 1}} {2}(k - \tfrac{2} {{k - 1}})$ . We also show that our result implies a simple short proof of Grotzsch's Theorem that every triangle-free planar graph is 3-colorable.

An upper bound on the number of high-dimensional permutations

Abstract: What is the higher-dimensional analog of a permutation? If we think of a permutation as given by a permutation matrix, then the following definition suggests itself: A d-dimensional permutation of order n is an n×n×...×n=[n] d+1 array of zeros and ones in which every line contains a unique 1 entry. A line here is a set of entries of the form {(x 1,...,x i?1,y,x i+1,...,x d+1)|n?y?1} for some index d+1?i?1 and some choice of x j ? [n] for all j ? i. It is easy to observe that a one-dimensional permutation is simply a permutation matrix and that a two-dimensional permutation is synonymous with an order-n Latin square. We seek an estimate for the number of d-dimensional permutations. Our main result is the following upper bound on their number $$\left( {(1 + o(1))\frac{n} {{e^d }}} \right)^{n^d } .$$ We tend to believe that this is actually the correct number, but the problem of proving the complementary lower bound remains open. Our main tool is an adaptation of Bregman's [1] proof of the Minc conjecture on permanents. More concretely, our approach is very close in spirit to Schrijver's [11] and Radhakrishnan's [10] proofs of Bregman's theorem.

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

Abstract: We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sarkozy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2?(G) monochromatic cycles, where ?(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least $$\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}$$ can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that $$\bar G$$ does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| ? c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C 4)=1, which is best possible.

Coloring intersection graphs of x-monotone curves in the plane

Abstract: A class of graphs G is ?-bounded if the chromatic number of the graphs in G is bounded by some function of their clique number. We show that the class of intersection graphs of simple families of x-monotone curves in the plane intersecting a vertical line is ?-bounded. As a corollary, we show that the class of intersection graphs of rays in the plane is ?-bounded, and the class of intersection graphs of unit segments in the plane is ?-bounded.

Regular graphs of large girth and arbitrary degree

Abstract: For every integer d?10, we construct infinite families {G n } n?? of d+1-regular graphs which have a large girth ?log d |G n |, and for d large enough ?1.33 · log d |G n |. These are Cayley graphs on PGL 2(F q ) for a special set of d+1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is a prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I n } n?? of d + 1-regular graphs, realized as Cayley graphs on SL 2(F q ), and which are displaying a girth ?0.48·log d |I n |. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M n } n?N of 2 k +1-regular graphs were shown to have girth ?2/3·log2 k |M n |.

Cocliques in the Kneser graph on the point-hyperplane flags of a projective space

Abstract: We prove an Erd?s-Ko-Rado-type theorem for the Kneser graph on the point-hyperplane flag in a finite projective space.

Three-point configurations determined by subsets of $$\mathbb{F}_q^2$$ via the Elekes-Sharir Paradigm

Abstract: We prove that if $$E \subset \mathbb{F}_Q^2$$ , q ? 3 mod 4, has size greater than $$Cq^{\tfrac{7} {4}}$$ , then E determines a positive proportion of all congruence classes of triangles in $$\mathbb{F}_q^2$$ . The approach in this paper is based on the approach to the Erd?s distance problem in the plane due to Elekes and Sharir, followed by an incidence bound for points and lines in $$\mathbb{F}_q^3$$ . We also establish a weak lower bound for a related problem in the sense that any subset E of $$\mathbb{F}_q^2$$ of size less than cq 4/3 definitely does not contain a positive proportion of translation classes of triangles in the plane. This result is a special case of a result established for n-simplices in $$\mathbb{F}_q^d$$ . Finally, a necessary and sufficient condition on the lengths of a triangle for it to exist in $$\mathbb{F}^2$$ for any field $$\mathbb{F}$$ of characteristic not equal to 2 is established as a special case of a result for d-simplices in $$\mathbb{F}^d$$ .

Sparsely intersecting perfect matchings in cubic graphs

Abstract: In 1971, Fulkerson made a conjecture that every bridgeless cubic graph contains a family of six perfect matchings such that each edge belongs to exactly two of them; equivalently, such that no three of the matchings have an edge in common. In 1994, Fan and Raspaud proposed a weaker conjecture which requires only three perfect matchings with no edge in common. In this paper we discuss these and other related conjectures and make a step towards Fulkerson's conjecture by proving the following result: Every bridgeless cubic graph which has a 2-factor with at most two odd circuits contains three perfect matchings with no edge in common.

Lower bounds for boxicity

Abstract: An axis-parallel b-dimensional box is a Cartesian product R1×–2×...×Rb where Ri is a closed interval of the form [ai; bi] on the real line. For a graph G, its boxicity box(G) is the minimum dimension b, such that G is representable as the intersection graph of boxes in b-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below: 1. The boxicity of a graph on n vertices with no universal vertices and minimum degree δ is at least n/2(n−δ−1). 2. Consider the G(n;p) model of random graphs. Let p ≤ 1 − 40logn/n2. Then with high probability, box(G) = Ω(np(1 − p)). On setting p = 1/2 we immediately infer that almost all graphs have boxicity Ω(n). Another consequence of this result is as follows: For any positive constant c < 1, almost all graphs on n vertices and \(m \leqslant c\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) edges have boxicity Ω(m/n). 3. Let G be a connected k-regular graph on n vertices. Let λ be the second largest eigenvalue in absolute value of the adjacency matrix of G. Then, the boxicity of G is at least \(\left( {\frac{{k^2 /\lambda ^2 }} {{\log \left( {1 + k^2 /\lambda ^2 } \right)}}} \right)\left( {\frac{{n - k - 1}} {{2n}}} \right)\). 4. For any positive constant c < 1, almost all balanced bipartite graphs on 2n vertices and m≤cn2 edges have boxicity Ω(m/n).

Constructing Ramsey graphs from Boolean function representations

Abstract: Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdos' probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2 n , the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1} n except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.

Optimal covers with Hamilton cycles in random graphs

Abstract: A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G n,p a.a.s. has size ??(G n,p )/2?. Glebov, Krivelevich and Szabo recently initiated research on the `dual' problem, where one asks for a set of Hamilton cycles covering all edges of G. Our main result states that for $$\tfrac{{log^{117} n}} {n} \leqslant p \leqslant 1 - n^{ - 1/8}$$ , a.a.s. the edges of G n,p can be covered by ?Δ (G n,p )/2? Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szabo, which holds for p ? n ?1+?. Our proof is based on a result of Knox, Kuhn and Osthus on packing Hamilton cycles in pseudorandom graphs.

Approximating minimum-cost edge-covers of crossing biset-families

Abstract: Part of this paper appeared in the preliminary version [16]. An ordered pair Ŝ = (S, S +) of subsets of a groundset V is called a biset if S ⊆ S+; (V S +;V S) is the co-biset of Ŝ. Two bisets $\hat X,\hat Y$ intersect if X X ? Y ? $ ot 0$ and cross if both X ∩ Y $ ot 0$ and X + ∪ Y + ≠= V. The intersection and the union of two bisets $\hat X,\hat Y$ are defined by $\hat X \cap \hat Y = (X \cap Y,X^ + \cap Y^ + )$ and $\hat X \cup \hat Y = (X \cup Y,X^ + \cup Y^ + )$ . A biset-family $\mathcal{F}$ is crossing (intersecting) if $\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}$ for any $\hat X,\hat Y \in \mathcal{F}$ that cross (intersect). A directed edge covers a biset Ŝ if it goes from S to V S +. We consider the problem of covering a crossing biset-family $\mathcal{F}$ by a minimum-cost set of directed edges. While for intersecting $\mathcal{F}$ , a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing $\mathcal{F}$ is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently or is not known. Let us say that a biset-family $\mathcal{F}$ is k-regular if $\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}$ for any $\hat X,\hat Y \in \mathcal{F}$ with |V (X∪Y)≥k+1 that intersect. In this paper we obtain an O(log |V|)-approximation algorithm for arbitrary crossing $\mathcal{F}$ if in addition both $\mathcal{F}$ and the family of co-bisets of $\mathcal{F}$ are k-regular, our ratios are: $O\left( {\log \frac{{|V|}} {{|V| - k}}} \right) $ if |S + \ S| = k for all $\hat S \in \mathcal{F}$ , and $O\left( {\frac{{|V|}} {{|V| - k}}\log \frac{{|V|}} {{|V| - k}}} \right) $ if |S + \ S| = k for all $\hat S \in \mathcal{F}$ . Using these generic algorithms, we derive for some network design problems the following approximation ratios: $O\left( {\log k \cdot \log \tfrac{n} {{n - k}}} \right) $ for k-Connected Subgraph, and O(logk) $\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\} $ for Subset k-Connected Subgraph when all edges with positive cost have their endnodes in the subset.

Steiner transitive-closure spanners of low-dimensional posets

Abstract: Given a directed graph G=(V, E) and an integer k ≥ 1, a k-transitive-closure spanner (k-TC-spanner) of G is a directed graph H=(V, E H ) that has (1) the same transitive closure as G and (2) diameter at most k. In some applications, the shortcut paths added to the graph in order to obtain small diameter can use Steiner vertices, that is, vertices not in the original graph G. The resulting spanner is called a Steiner transitive-closure spanner (Steiner TC-spanner). Motivated by applications to property reconstruction and access control hierarchies, we concentrate on Steiner TC-spanners of directed acyclic graphs or, equivalently, partially ordered sets. In these applications, the goal is to find a sparsest Steiner k-TC-spanner of a poset G for a given k and G. The focus of this paper is the relationship between the dimension of a poset and the size of its sparsest Steiner TC-spanner. The dimension of a poset G is the smallest d such that G can be embedded into a d-dimensional directed hypergrid via an order-preserving embedding. We present a nearly tight lower bound on the size of Steiner 2-TC-spanners of d- dimensional directed hypergrids. It implies better lower bounds on the complexity of local reconstructors of monotone functions and functions with small Lipschitz constant. The lower bound is derived from an explicit dual solution to a linear programming relaxation of the Steiner 2-TC-spanner problem. We also give an efficient construction of Steiner 2-TC-spanners, of size matching the lower bound, for all low-dimensional posets. Finally, we present a lower bound on the size of Steiner k-TC-spanners of d-dimensional posets. It shows that the best-known construction, due to De Santis et al., cannot be improved significantly.

On the minimal fourier degree of symmetric Boolean functions

Abstract: In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function Specifically, we prove that for every non-linear and symmetric f: {0, 1} k → {0, 1} there exists a set; $ ot 0 e S \subset [k]$ such that ¦S¦ = O(Γ(k)+√k, and $\hat f(S) e 0$ where Γ(m)≤m 0525 is the largest gap between consecutive prime numbers in {1,, m} As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al [10] Our bound on the degree is a significant improvement over the previous result of Kolountzakis et al [8] who proved that ¦S¦=O(k=log k) We also show a connection between lower-bounding the degree of non-constant functions that take values in {0,1,2} and the question that we study here

Judicious partitions of uniform hypergraphs

Abstract: The vertices of any graph with m edges may be partitioned into two parts so that each part meets at least $$\tfrac{{2m}} {3}$$ edges. Bollobas and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges may likewise be partitioned into r classes such that each part meets at least $$\tfrac{r} {{2r - 1}}$$ edges. In this paper we prove the weaker statement that, for each r ? 4, a partition into r classes may be found in which each class meets at least $$\tfrac{r} {{3r - 4}}$$ edges, a substantial improvement on previous bounds.

Triangulations of the sphere, bitrades and abelian groups

Abstract: Let $$\mathcal{G}$$ be a triangulation of the sphere with vertex set V, such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined $$\mathcal{A}_W$$ to be the abelian group generated by the set V, with relations r+c+s = 0 for all white triangles with vertices r, c and s. The group $$\mathcal{A}_B$$ can be de fined similarly, using black triangles. The paper shows that $$\mathcal{A}_W$$ and $$\mathcal{A}_B$$ are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the structure of $$\mathcal{A}_W$$ and the theory of asymmetric Laplacians of finite directed graphs, and weaker results for orientable surfaces of higher genus are given. The relevance of the group $$\mathcal{A}_W$$ to the understanding of the embeddings of a partial latin square in an abelian group is also explained.

Edge-transitive dihedral or cyclic covers of cubic symmetric graphs of order 2P

Abstract: A regular cover of a connected graph is called dihedral or cyclic if its transformation group is dihedral or cyclic, respectively. Let X be a connected cubic symmetric graph of order 2p for a prime p. Several publications have investigated the classification of edge-transitive dihedral or cyclic covers of X for specific p. The edge-transitive dihedral covers of X have been classified for p=2 and the edge-transitive cyclic covers of X have been classified for p≤5. In this paper an extension of the above results to an arbitrary prime p is presented.

Measurable events indexed by products of trees

Abstract: A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ? 2, called the branching number of T, such that every t ? T has exactly b immediate successors. A vector homogeneous tree T is a finite sequence (T 1,...,T d ) of homogeneous trees and its level product ?T is the subset of the Cartesian product T 1×...×T d consisting of all finite sequences (t 1,...,t d ) of nodes having common length. We study the behavior of measurable events in probability spaces indexed by the level product ?T of a vector homogeneous tree T. We show that, by refining the index set to the level product ?S of a vector strong subtree S of T, such families of events become highly correlated. An analogue of Lebesgue's density Theorem is also established which can be considered as the "probabilistic" version of the density Halpern-Lauchli Theorem.

Plünnecke and Kneser type theorems for dimension estimates

Abstract: Given a division ring K containing the field k in its center and two finite subsets A and B of K*, we give some analogues of Plunnecke and Kneser Theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. We also explain how they imply the corresponding more classical theorems for abelian groups. These Plunnecke type estimates are then generalized to the case of associative algebras. We also obtain an analogue in the context of division rings of a theorem by Tao describing the sets of small doubling in a group.

Idiosynchromatic Poetry

Abstract: We prove equivalences of asymmetric partition relations involving natural numbers and products of weakly compact cardinals ?, infinite cardinals ?

Strong approximation in random towers of graphs

Abstract: Let T=T 2 be the rooted binary tree, Aut(T) = $\mathop {\lim }\limits_ \leftarrow $ Aut n (T) its automorphism group and ? n : Aut(T)?Aut n (T) the restriction maps to the first n levels of the tree. If L n is the the n th level of the tree then Aut n (T) < Sym(L n ) can be identified with the 2-Sylow subgroup of the symmetric group on 2 n points. Consider a random subgroup Γ:= 〈a〉= 〈a 1, a 2,..., a m 〉 ∈ Aut(T) m generated by m independent Haar-random tree automorphisms. Theorem A. The following hold, almost surely, for every non-cyclic subgroup Δ < Γ:

Connected Baranyai's theorem

Abstract: Let K n h = (V, ( h V )) be the complete h-uniform hypergraph on vertex set V with ¦V¦ = n. Baranyai showed that K n h can be expressed as the union of edge-disjoint r-regular factors if and only if h divides rn and r divides $(_{h - 1}^{n - 1} )$ . Using a new proof technique, in this paper we prove that ?K n h can be expressed as the union $\mathcal{G}_1 \cup ... \cup \mathcal{G}_k $ of k edge-disjoint factors, where for 1≤i≤k, $\mathcal{G}_i $ is r i -regular, if and only if (i) h divides r i n for 1≤i≤k, and (ii) $\sum olimits_{i = 1}^k {r_i = \lambda (_{h - 1}^{n - 1} )} $ . Moreover, for any i (1≤i≤k) for which r i ?2, this new technique allows us to guarantee that $\mathcal{G}_i $ is connected, generalizing Baranyai's theorem, and answering a question by Katona.

Extremal results for odd cycles in sparse pseudorandom graphs

Abstract: We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and ? the generalized Turan density ? F (?) denotes the relative density of a maximum subgraph of ?, which contains no copy of F. Extending classical Turan type results for odd cycles, we show that ? F (?)=1/2 provided F is an odd cycle and ? is a sufficiently pseudorandom graph. In particular, for (n,d,?)-graphs ?, i.e., n-vertex, d-regular graphs with all non-trivial eigenvalues in the interval [??,?], our result holds for odd cycles of length l, provided $$\lambda ^{\ell - 2} \ll \frac{{d^{\ell - 1} }} {n}\log (n)^{ - (\ell - 2)(\ell - 3)} .$$ Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szabo, and Vu, who addressed the case when F is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free (n,d;?)-graphs) shows that our assumption on ? is best possible up to the polylog-factor for every odd l?5.

A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Abstract: Let G be a connected simple graph, let X⊆V (G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyarfas, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d T (x) for all x ? X, where d T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d T (x) ? f(x) for all x ? X if and only if for any nonempty subset S ⊆ X, |N G (S) ? S| ? f(S) + 2|S| ? ? G (S) ?, where ? G (S) is the number of components of the subgraph induced by S.

A nullstellensatz for sequences over $\mathbb{F}_p $

Abstract: Let p be a prime and let A = (a 1,...,a l ) be a sequence of nonzero elements in $\mathbb{F}_p $ . In this paper, we study the set of all 0---1 solutions to the equation $$a_1 x_1 + \cdots + a_\ell x_\ell = 0$$ We prove that whenever l?p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l

The circumference of the square of a connected graph

Abstract: The celebrated result of Fleischner states that the square of every 2-connected graph is Hamiltonian. We investigate what happens if the graph is just connected. For every n ? 3, we determine the smallest length c(n) of a longest cycle in the square of a connected graph of order n and show that c(n) is a logarithmic function in n. Furthermore, for every c ? 3, we characterize the connected graphs of largest order whose square contains no cycle of length at least c.