Complete Minors In <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">s,s</Emphasis></Subscript>-Free Graphs

We prove that for a fixed integer s2 every K _s,s -free graph of average degree at least r contains a K _p minor where . A well-known conjecture on the existence of dense K _s,s -free graphs would imply that the value of the exponent is best possible. Our result implies Hadwigers conjecture for K _s,s -free graphs whose chromatic number is sufficiently large compared with s.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">1</Emphasis></Subscript> of a self-product of a curve

We present elements of H¹(C×C, ᵊ6₂) for certain specific curves C. The image of the element under the boundary map arising from the localization sequence of K-theory is the graph of frobenius endomorphism of the reduction of the curve modulo 3.     

Graded central polynomials for the algebra <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let M _n (K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ _n -grading and a natural ℤ-grading. Finite bases for its ℤ _n -graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ _n -graded and for the ℤ-graded central polynomials for M _n (K) Partially supported by CNPq 620025/2006-9     

On hamiltonicity of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-dominated graphs

We introduce a new class of graphs which we call P ₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P ₃-dominated graph. We prove that G is hamiltonian if α(G ²) ≤ κ(G), with two exceptions: K _2,3 and K _1,1,3. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. This paper was completed when both authors visited the Center for Combinatorics, Nankai University, Tianjin. They gratefully acknowledge the hospitality and support of the Center for Combinatorics and Nankai University. The work of E.Vumar is sponsored by SRF for ROCS, REM.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

Induced Subdivisions In <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">s,s</Emphasis></Subscript>-Free Graphs of Large Average Degree

We prove that for every graph H and for every s there exists d=d(H,s) such that every graph of average degree at least d contains either a K_s,s as a subgraph or an induced subdivision of H.     

Uniquely <Emphasis Type="Italic">C</Emphasis><Subscript>4</Subscript>-Saturated Graphs

For a fixed graph H, a graph G is uniquely H-saturated if G does not contain H, but the addition of any edge from [`(G)]{\overline{G}} to G completes exactly one copy of H. Using a combination of algebraic methods and counting arguments, we determine all the uniquely C ₄-saturated graphs; there are only ten of them.     

The -conjecture and equivariant <Emphasis Type="Italic">e</Emphasis><Subscript><Emphasis Type="Italic">C</Emphasis></Subscript>-invariants

Let X be a smooth closed oriented non-spin 4-manifold with even intersection form kE₈nH (n1). The -conjecture states that n is greater than or equal to |k|. In this paper we give a proof of the -conjecture. The strategy of this paper is to use the finite dimensional approximation of the map induced from the Seiberg-Witten equations and equivariant e_C-invariants as in the paper of M. Furuta and Y. Kametani.Mathematics Subject Classification (1991): 57R55This work was supported by Korea Research Foundation Grant (KRF–2002–003–C00011).     

<Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free subgraphs of random graphs revisited

In Combinatorica 17(2), 1997, Kohayakawa, ?uczak and Rödl state a conjecture which has several implications for random graphs. If the conjecture is true, then, for example, an application of a version of Szemerédi’s regularity lemma for sparse graphs yields an estimation of the maximal number of edges in an H-free subgraph of a random graph G _{n, p} . In fact, the conjecture may be seen as a probabilistic embedding lemma for partitions guaranteed by a version of Szemerédi’s regularity lemma for sparse graphs. In this paper we verify the conjecture for H = K ₄, thereby providing a conceptually simple proof for the main result in the paper cited above.     

On the Chromatic Number of the Square of the Kneser Graph <Emphasis Type="Italic">K</Emphasis>(2<Emphasis Type="Italic">k</Emphasis>+1, <Emphasis Type="Italic">k</Emphasis>)

The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The chromatic number of the Kneser graph K(n, k) is n–2k+2. Zoltán Füredi raised the question of determining the chromatic number of the square of the Kneser graph, where the square of a graph is the graph obtained by adding edges joining vertices at distance at most 2. We prove that (K²(2k+1, k))4k when k is odd and (K²(2k+1, k))4k+2 when k is even. Also, we use intersecting families of sets to prove lower bounds on (K²(2k+1, k)), and we find the exact maximum size of an intersecting family of 4-sets in a 9-element set such that no two members of the family share three elements.This work was partially supported by NSF grant DMS-0099608Final version received: April 23, 2003     

On the b-Coloring of Cographs and <Emphasis Type="Italic">P</Emphasis><Subscript>4</Subscript>-Sparse Graphs

A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromatic number of a graph G, denoted by χ _b (G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is b-continuous if it admits a b-coloring with t colors, for every . We define a graph G to be b-monotonic if χ _b (H ₁) ≥ χ _b (H ₂) for every induced subgraph H ₁ of G, and every induced subgraph H ₂ of H ₁. In this work, we prove that P ₄-sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic. Besides, we describe a dynamic programming algorithm to compute the b-chromatic number in polynomial time within these graph classes. Flavia Bonomo: Partially supported by ANPCyT PICT-2007-00533 and PICT-2007-00518, and UBACyT Grants X069 and X606 (Argentina). Guillermo Durán: Partially supported by FONDECyT Grant 1080286 and Millennium Science Institute “Complex Engineering Systems” (Chile), and ANPCyT PICT-2007-00518 and UBACyT Grant X069 (Argentina). Javier Marenco: Partially supported by ANPCyT PICT-2007-00518 and UBACyT Grant X069 (Argentina).     

Exponentially Many Monochromatic <Emphasis Type="Italic">n</Emphasis>-Matchings in <Emphasis Type="Italic">K</Emphasis><Subscript>3<Emphasis Type="Italic">n</Emphasis>-1</Subscript>

Erdös et al and Gerencsér et al had shown that in any 2-edge-coloring of K _3n-1, there is a n-matching containing edges with the same color(we call such matching monochromatic matching). In this paper we show that for any 2-edge-coloring of K _3n-1 there exists a monochromatic subgraph H of K _3n-1 which contains exponentially many monochromatic n-matchings.     

Maximal regular subsemibands of finite order-preserving transformation semigroups <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">r</Emphasis>)

Let O _n be the order-preserving transformation semigroup on X _n . For an arbitrary integer r such that 1≤r≤n−2, we completely describe the maximal regular subsemibands of the semigroup K(n,r)={α∈O _n :|im(α)|≤r}. We also formulate the cardinal number of such subsemigroups.     

The Combinatorial Quantum Cohomology Ring of <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">B</Emphasis>

A purely combinatorial construction of the quantum cohomology ring of the generalized flag manifold is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of our previous papers [12, 13], we obtain a presentation of this ring in terms of generators and relations, and formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which—for G = SL_n( )—was proved previously in the paper [5].     

On Resolvable Multipartite <Emphasis Type="Italic">G</Emphasis>-Designs

A necessary and sufficient condition for the existence of a _km–factorization of the complete symmetric k–partite multi-digraph K*(n₁,n₂,...,n_k) is obtained for odd k. As a consequence, a resolvable (k,n,km,) multipartite _km–design exists for odd k if and only if m|n. This deduces a result of Ushio when m=1 and k=3. Further, a necessary and sufficient condition for the existence of a _km–factorization of is established for even k, where denotes the wreath product of graphs. Finally, a simple and short proof for the non-existence of a _k–factorization of is obtained for odd k.Acknowledgments.The author thanks Dr. P. Paulraja for his useful ideas in writing this paper and the Department of Science and Technology, New Delhi, for its support (Project Grant No. DST/MS/103/99).Final version received: November 17, 2003     

Note making a <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free graph bipartite

We show that every K ₄-free graph G with n vertices can be made bipartite by deleting at most n ²/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

On <Emphasis Type="Italic">K</Emphasis><Subscript>1,3</Subscript>-free strictly Deza graphs

A graph on v vertices is called a Deza graph with parameters (v, k, b, a) if it is k-regular and the number of common neighbors of two distinct vertices takes on one of two values. We describe strictly Deza graphs that do not contain K _1,3 among their induced subgraphs and are unions of closed neighborhoods of two nonadjacent vertices. The latter condition means that there are two nonadjacent vertices such that any other vertex is adjacent to at least one of the them.     

Homology of GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>: injectivity conjecture for GL<Subscript>4</Subscript>

The homology of GL _n (R) and SL _n (R) is studied, where R is a commutative ‘ring with many units’. Our main theorem states that the natural map H ₄(GL₃(R), k) → H ₄(GL₄(R), k) is injective, where k is a field with char(k) ≠ 2, 3. For an algebraically closed field F, we prove a better result, namely, is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K ₄(R).     

<Emphasis Type="Italic">S</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis>,exp</Subscript> Does Not Prove NP = <Emphasis Type="Italic">co</Emphasis>-NP Uniformly

The notion of a uniform sequent-calculus proof is introduced. It is shown that a strengthening, S_k,exp , of the well-studied bounded arithmetic system S_k of Buss does not prove NP = co-NP with a uniform proof. A slightly stronger result that S_k,exp cannot prove uniformly for 2 ≤ k′ ≤ k is also established. A modification of the technique used is applied to show that S_k,exp is unable to prove the Davis-Putnam-Robinson-Matiyasevich theorem. Generalizations of these results to higher levels of the Grzegorczyck hierarchy are presented. Bibliography: 21 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 304, 2003, pp. 99–120.