An upper bound for the number of matroids

An upper bound for the number of matroids is obtained. This upper bound complements the lower bound obtained by Piff and Welsh in [1].     

An upper bound for the total chromatic number

The total chromatic number,(G), of a graphG, is defined to be the minimum number of colours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that , where(G) is the vertex chromatic number and(G) is the edge chromatic number of the graph.Partially supported by ORS grant ORS/84120     

An upper bound for the restricted online Ramsey number

The restricted $(m,n;N)$-online Ramsey game is a game played between two players, Builder and Painter. The game starts with $N$ isolated vertices. Each turn Builder picks an edge to build and Painter chooses whether that edge is red or blue, and Builder aims to create a red $K_m$ or blue $K_n$ in as few turns as possible. The restricted online Ramsey number $\stackrel{?}{r}(m,n;N)$ is the minimum number of turns that Builder needs to guarantee her win in the restricted $(m,n;N)$-online Ramsey game. We show that if $N=r(n,n)$, $\stackrel{?}{r}(n,n;N)\le \left(\genfrac{}{}{0ex}{}{N}{2}\right)?\Omega (NlogN),$motivated by a question posed by Conlon, Fox, Grinshpun and He. The equivalent game played on infinitely many vertices is called the online Ramsey game. As almost all known Builder strategies in the online Ramsey game end up reducing to the restricted setting, we expect further progress on the restricted online Ramsey game to have applications in the general case.     

An upper bound on the number of planarK-sets

Given a setS ofn points, a subsetX of sizek is called ak-set if there is a hyperplane that separatesX fromS–X. We prove thatO(nk/log*k) is an upper bound for the number ofk-sets in the plane, thus improving the previous bound of Erdös, Lovász, Simmons, and Strauss by a factor of log*k.The research of J. Pach was supported in part by NSF Grant CCR-8901484 and by Grant OTKA-1418 from the Hungarian Foundation for Scientific Research. The research of W. Steiger and E. Szemerédi was supported in part by NSF Grant CCR-8902522. All authors express gratitude to the NSF DIMACS Center at Rutgers.     

An upper bound for the k-domination number of a graph

The k-domination number of a graph G, γk(G), is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k, then γk(G) ≤ kp/(k + 1).     

An upper bound for the chromatic number of line graphs

It was conjectured by Reed [B. Reed, ω,α, and χ, Journal of Graph Theory 27 (1998) 177–212] that for any graph G, the graph’s chromatic number χ(G) is bounded above by , where Δ(G) and ω(G) are the maximum degree and clique number of G, respectively. In this paper we prove that this bound holds if G is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph G and produces a colouring that achieves our bound.     

An upper bound for the path number of a graph

The path number of a graph G, denoted p(G), is the minimum number of edge-disjoint paths covering the edges of G. Lovász has proved that if G has u odd vertices and g even vertices, then p(G) ≤ 1/2 u + g - 1 ≤ n - 1, where n is the total number of vertices of G. This paper clears up an error in Lovász's proof of the above result and uses an extension of his construction to show that p(G) ≤ 1/2 u + [3/4g] ≤ [3/4n].     

An upper bound on the number of high-dimensional permutations

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 Brégman’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 Brégman’s theorem.     

On the upper bound of uniformly bounded solutions

New conditions of uniform boundedness of solutions are obtained and methods for calculating upper bounds are suggested.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 501–505, May, 1994.     

An upper bound on the number of frequency hypercubes

A frequency n-cube $F^n(q;l_0,...,l_{m-1})$ is an n-dimensional q-by-...-by-q array, where $q=l_0+...+l_{m-1}$, filled by numbers $0,...,m-1$ with the property that each line contains exactly $l_i$ cells with symbol i, $i=0,...,m-1$ (a line consists of q cells of the array differing in one coordinate). The trivial upper bound on the number of frequency n-cubes is $m^{{(q-1)}^n}$. We improve that lower bound for $n>2$, replacing $q-1$ by a smaller value s, by constructing a testing set of size $s^n$ for frequency n-cubes (a testing set is a collection of cells of an array the values in which uniquely determine the array with given parameters). We also construct new testing sets for generalized frequency n-cubes, which are essentially correlation-immune functions in n q-valued arguments; the cardinalities of new testing sets are smaller than for testing sets known before.     

An upper bound for the total chromatic number of dense graphs

In this paper we consider those graphs that have maximum degree at least 1/k times their order, where k is a (small) positive integer. A result of Hajnal and Szemerédi concerning equitable vertex-colorings and an adaptation of the standard proof of Vizing's Theorem are used to show that if the maximum degree of a graph G satisfies Δ(G) ≥ |V(G)/k, then X″(G) ≤ Δ(G) + 2k + 1. This upper bound is an improvement on the currently available upper bounds for dense graphs having large order.     

An upper bound for the harmonious chromatic number of a graph

An upper bound for the harmonious chromatic number of a graph G is given. Three corollaries of the theorem are theorems or improvements of the theorems of Miller and Pritikin.     

An upper bound for domination number of 5-regular graphs

Let G = (V, E) be a simple graph. A subset S ⊆ V is a dominating set of G, if for any vertex u ∈ V-S, there exists a vertex v ∈ S such that uv ∈ E. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. In this paper we will prove that if G is a 5-regular graph, then γ(G) ⩽ 5/14n.     

An upper bound for the minimum weight of dual codes of Figueroa planes

In this note we give an explicit construction for words of weight 2q³ - q² - q in the dual p-ary code of the Figueroa plane of order q³, where q > 2 is any power of the prime p. When p is odd this then allows us, for the Figueroa planes, to improve on the previously known upper bound of 2q³ for the minimum weight of the dual p-ary code of any plane of order q³. The construction is the same as one that applies to desarguesian planes of order q³ as described in [3].     

Asymptotic bound on binary self-orthogonal codes

We present two constructions for binary self-orthogonal codes. It turns out that our constructions yield a constructive bound on binary self-orthogonal codes. In particular, when the information rate R = 1/2, by our constructive lower bound, the relative minimum distance δ ≈ 0.0595 (for GV bound, δ ≈ 0.110). Moreover, we have proved that the binary self-orthogonal codes asymptotically achieve the Gilbert-Varshamov bound. This work was supported by the China Scholarship Council, National Natural Science Foundation of China (Grant No.10571026), the Cultivation Fund of the Key Scientific and Technical Innovation Project of Ministry of Education of China, and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20060286006)