The order dimension of two levels of the Boolean lattice

LetB _n(s, t) denote the partially ordered set consisting of alls-subsets andt-subsets of ann-element underlying set where these sets are ordered by inclusion. Answering a question of Trotter we prove that dim(B _n(k, n–k))=n–2 for 3k<(1/7)n ^1/3. The proof uses extremal hypergraph theory.     

An extremal set theoretical characterization of some steiner systems

Letn, k, t be integers,n>k>t≧0, and letm(n, k, t) denote the maximum number of sets, in a family ofk-subsets of ann-set, no two of which intersect in exactlyt elements. The problem of determiningm(n, k, t) was raised by Erdős in 1975. In the present paper we prove that ifk≦2t+1 andk−t is a prime, thenm(n, k, t)≦( _t ⁿ )( _k ^2k-t-1 )/( _t ^2k-t-1 ). Moreover, equality holds if and only if an (n, 2k−t−1,t)-Steiner system exists. The proof uses a linear algebraic approach.     

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.     

The Rényi–Ulam pathological liar game with a fixed number of lies

The q-round Rényi–Ulam pathological liar game with k lies on the set [n]{1,…,n} is a 2-player perfect information zero sum game. In each round Paul chooses a subset A[n] and Carole either assigns 1 lie to each element of A or to each element of [n]A. Paul wins if after q rounds there is at least one element with k or fewer lies. The game is dual to the original Rényi–Ulam liar game for which the winning condition is that at most one element has k or fewer lies. Define to be the minimum n such that Paul can win the q-round pathological liar game with k lies and initial set [n]. For fixed k we prove that is within an absolute constant (depending only on k) of the sphere bound, ; this is already known to hold for the original Rényi–Ulam liar game due to a result of J. Spencer.     

The game chromatic number of random graphs

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number χ_g(G) is the minimum k for which the first player has a winning strategy. In this study, we analyze the asymptotic behavior of this parameter for a random graph G_n,p. We show that with high probability, the game chromatic number of G_n,p is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Solutions of the matrix equation (1):X m+A 1Xm−1+…+A m=0

SupposeX and the coefficientsA ₁, …,A _m aren×n matrices. LetB be anmn×mn matrix as in (7). LetJ be the Jordan canonical matrix ofB andB=PJP ⁻. LetE _i denote thei×i unit matrix.V is defined bydV/dt=JV andV(t=0)=E _mn. ThenZ=PV satisfiesdZ/dt=BZ.P ^* is a matrix which consists of the firstn rows ofP. The author proves: There is a solution of (1) ↔ there are anmn×n matrixC, ann×n matrixQ and ann×n function matrixN such thatP ^*VC=QN, where detQ≠0 andN is defined byN(t=0)=E _n anddN/dt=RN, in whichR is ann×n Jordan canonical matrix. There are three cases regarding the solutions of (1): No solution, finitek solutions, 1<k<C _n ^m , and infinite solutions which containj parameters, 1<-j<-mn ².     

Computing solutions for matching games

A matching game is a cooperative game (N, v) defined on a graph G = (N, E) with an edge weighting w: E? \mathbb R₊{w: E\to {\mathbb R}_+}. The player set is N and the value of a coalition S í N{S \subseteq N} is defined as the maximum weight of a matching in the subgraph induced by S. First we present an O(nm + n ² log n) algorithm that tests if the core of a matching game defined on a weighted graph with n vertices and m edges is nonempty and that computes a core member if the core is nonempty. This algorithm improves previous work based on the ellipsoid method and can also be used to compute stable solutions for instances of the stable roommates problem with payments. Second we show that the nucleolus of an n-player matching game with a nonempty core can be computed in O(n ⁴) time. This generalizes the corresponding result of Solymosi and Raghavan for assignment games. Third we prove that is NP-hard to determine an imputation with minimum number of blocking pairs, even for matching games with unit edge weights, whereas the problem of determining an imputation with minimum total blocking value is shown to be polynomial-time solvable for general matching games.     

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.     

Analyzing n-player impartial games

Combinatorial game theory is the study of two player perfect information games. While work has been done in the past on expanding this field to include n-player games we present a unique method which guarantees a single winner. Specifically our goal is to derive a function which, given an n-player game, is able to determine the winning player (assuming all n players play optimally). Once this is accomplished we use this function in analyzing a certain family of three player subtraction games along with a complete analysis of three player, three row Chomp. Furthermore we make use of our new function in producing alternative proofs to various well known two player Chomp games. Finally the paper presents a possible method of analyzing a two player game where one of the players plays a completely random game. As it turns out this slight twist to the rules of combinatorial game theory produces rather interesting results and is certainly worth the time to study further.     

Augmenting Edge-Connectivity over the Entire Range inÕ(nm) Time

For a given undirected graphG = (V, E, c_G) with edges weighted by nonnegative realsc_G:E → R ⁺ , let Λ_G(k) stand for the minimum amount of weights which needs to be added to makeG k-edge-connected, and letG*(k) be the resulting graph obtained fromG. This paper first shows that function Λ_Gover the entire rangek [0, +∞] can be computed inO(nm + n² log n) time, and then shows that allG*(k) in the entire range can be obtained fromO(n log n) weighted cycles, and such cycles can be computed inO(nm + n² log n) time, wherenandmare the numbers of vertices and edges, respectively.     

Concerning the order of approximation of periodic continuous functions by trigonometric interpolation polynomials

Letx _kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR _n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R _n(x_kn)=f(x_kn),R _n ^(j)(x_kn)=0,j=1,2,4,k=0,1…,n−1. We setw ₂(t,f) as the second modulus of continuity off(x). Then we prove that |R _n(x)-f(x)|=0(nw₂(1/nf)). We also examine the question of lower estimate of ‖R _n-f‖. This generalizes an earlier work of the author.     

Achievement and avoidance games for generating abelian groups

For any finite groupG, the DO GENERATE game is played by two players Alpha and Beta as follows. Alpha moves first and choosesx ₁ ∈G. Thek-th play consists of a choice ofx _k ∈G ?S _k ?1 whereS _n ={itx ₁,...,x _n }. LetG _n = 〈S _n 〉. The game ends whenG _n =G. The player who movesx _n wins. In the corresponding avoidance game, DON'T GENERATE, the last player to move loses. Of course neither game can end in a draw. For an arbitrary group, it is an unsolved problem to determine whether Alpha or Beta wins either game. However these two questions are answered here for abelian groups.     

The relaxed game chromatic index of k-degenerate graphs

The (r,d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r,d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with Δ(G)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ+k-1 and d?2k²+4k.     

Game chromatic number of strong product graphs

The graph coloring game is a two-player game in which the two players properly color an uncolored vertex of G alternately. The first player wins the game if all vertices of G are colored, and the second wins otherwise. The game chromatic number of a graph G is the minimum integer k such that the first player has a winning strategy for the graph coloring game on G with k colors. There is a lot of literature on the game chromatic number of graph products, e.g., the Cartesian product and the lexicographic product. In this paper, we investigate the game chromatic number of the strong product of graphs, which is one of major graph products. In particular, we completely determine the game chromatic number of the strong product of a double star and a complete graph. Moreover, we estimate the game chromatic number of some King's graphs, which are the strong products of two paths.     

Absolute gradient bounds for nonparametric hypersurfaces of constant mean curvature

We consider the problem of determining the existence of absolute apriori gradient bounds of nonparametric hypersurfaces of constant mean curvature in ann-dimensional sphereB _R, 1>R>R ₀ ⁽ⁿ⁾ , (R ₀ ⁽ⁿ⁾ being a constant depending only onn), without imposing boundary conditions or bounds of any sort.

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Sunto Consideriamo il problema di determinare stime a priori di gradienti di ipersuperfici non parametriche di curvatura media costante in una sferan-dimensionaleB _R, 1>R>R ₀ ⁽ⁿ⁾, (R ₀ ⁽ⁿ⁾ essendo una costante che dipende solo dan), senza imporre condizioni al contorno o limiti di altro tipo.

     

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ₀, μ₁, ..., μ_s are distinct residues modp (p is a prime) such thatk ≡ μ₀ (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μ_i (modp) for somei, 1 ≦i≦s, then |ℱ|≦( _s ⁿ ). As a consequence we show that ifR ⁿ is covered bym sets withm<(1+o(1)) (1.2) ⁿ then there is one set within which all the distances are realised. It is left open whether the same conclusion holds for compositep.     

Tournament games and positive tournaments

Given a tournament T, the tournament game on T is as follows: Two players independently pick a node of T. If both pick the same node, the game is tied. Otherwise, the player whose node is at the tail of the arc connecting the two nodes wins. We show that the optimal mixed strategy for this game is unique and uses an odd number of nodes. A tournament is positive if the optimal strategy for its tournament game uses all of its nodes. The uniqueness of the optimal strategy then gives a new tournament decomposition: any tournament can be uniquely partitioned into positive subtournaments P₁, P₂, ?,P_k, so P_i “beats” P_j for all 1 ≤ i > j ≤ k. We count the number of n node positive tournaments and list them for n ≤ 7. © 1995 John Wiley & Sons, Inc.     

Geodetic contraction games on graphs

LetG = (V, E) be a finite graph. The setL of labeled vertices is initially empty. Two players A and B move alternately (A first), by choosing an unlabled vertexu V\L; thenu itself and all vertices on shortest paths betweenu and any vertex ofL are adjoined toL. WhenL =V, the game is over. Innormal play, the first player unable to move loses and his opponent wins. The outcome is reversed formisère play. We resolve the game by determining its winning strategies for the following cases: trees in normal play; cycles in normal and misère play; and complete graphsK _m with rays all of lengthn, in normal play.Partial support by Canadian Research Grants 69-0695 and A-4792 during visits at the University of Calgary and Simon Fraser University is gratefully acknowledged.     

A class of constructions for turán’s (3, 4)-problem

Letf(n) denote the minimal number of edges of a 3-uniform hypergraphG=(V, E) onn vertices such that for every quadrupleY ⊂V there existsY ⊃e ∈E. Turán conjectured thatf(3k)=k(k−1)(2k−1). We prove that if Turán’s conjecture is correct then there exist at least 2 ^k−2 non-isomorphic extremal hypergraphs on 3k vertices.     

GEODETIC GAMES FOR GRAPHS

ABSTRACT

Let S be a subset of the vertex set V(G) of a nontrivial connected graph G. The geodetic closure (S) of S is the set of all vertices on geodesics between two vertices in S. The first player A chooses a vertex v₁ of G. The second player B then picks v₂ ≠ v₁ and forms the geodetic closure (S₂) = ({v₁, v₂}). Now A selects v₃ ε V—S₂ and forms (S₃) = ({v₁, v₂, v₃}), etc. The player who first selects a vertex v_n such that (S_n) = V wins the achievement game, but loses the avoidance game. These geodetic achievement and avoidance games are solved for several families of graphs by determining which player is the winner.