Electronic Notes in Discrete Mathematics最新文献

The sum of random variables (errors) is the key element both for its statistical study and for the estimation and control of errors in many scientific and technical applications. In this paper we analyze the sum of independent random variables (independent errors) in spheres. This type of errors are very important, for example, in quantum computing. We prove that, given two independent isotropic random variables in an sphere, X₁ and X₂, the variance verifies $V(X_1+X_2)=V\left(X_1\right)+V\left(X_2\right)-\frac{V\left(X_1\right)V\left(X_2\right)}{2}$ and we conjecture that this formula is also true for non-isotropic random variables.

We investigate the average range of 1-Lipschitz mappings (graph-indexed random walks) of a given connected graph. This parameter originated in statistical physics, it is connected to the study of random graph homomorphisms and generalizes standard random walks on $Z$.

Our first goal is to prove a closed-form formula for this parameter for cycle graphs. The second one is to prove two conjectures, the first by Benjamini, Häggström and Mossel and the second by Loebl, Nešetřil and Reed, for unicyclic graphs. This extends a result of Wu, Xu, and Zhu [5] who proved the aforementioned conjectures for trees.

Segre's lemma of tangents dates back to the 1950's when he used it in the proof of his “arc is a conic” theorem. Since then it has been used as a tool to prove results about various objects including internal nuclei, Kakeya sets, sets with few odd secants and further results on arcs. Here, we survey some of these results and report on how re-formulations of Segre's lemma of tangents are leading to new results.

We prove that, for any $t\ge 3$, there exists a constant $c=c\left(t\right)>0$ such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d}^{t-1}/n^{t-2}$ contains $(1-o(1\left)\right)n/t$ vertex-disjoint copies of $K_t$. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), pp. 403–426] that ($n,d,\lambda$)-graphs with $n\in 3N$ and $\lambda \le c{d}^2$ for a suitably small absolute constant $c>0$ contain triangle-factors.

For integers k, n with $k,n\ge 1$, the n-color weak Schur number $W{S}_k\left(n\right)$ is defined as the least integer N, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution $x_1,\dots ,x_k,x_{k+1}$ in that interval to the equation $x_1+x_2+\dots +x_k=x_{k+1}$, with $x_i\ne x_j$, when $i\ne j$. We show a relationship between $W{S}_k(n+1)$ and $W{S}_k\left(n\right)$ and a general lower bound on the $W{S}_k\left(n\right)$ is obtained.

Let n and k be positive integers with $n\ge 2k$. Consider a circle C with n points $1,\dots ,n$ in clockwise order. The interlacing graph ${\mathrm{IG}}_{n,k}$ is the graph with vertices corresponding to k-subsets of [n] that do not contain two adjacent points on C, and edges between k-subsets P and Q if they interlace: after removing the points in P from C, the points in Q are in different connected components. In this paper we prove that the circular chromatic number of ${\mathrm{IG}}_{n,k}$ is equal to $n/k$, hence the chromatic number is $\lceil n/k\rceil$, and that its independence number is $\left(\begin{array}{c}n-k-1\\ k-1\end{array}\right)$.

We prove that for every convex subdivision of $R^d$ into n cells there exists a k-flat stabbing $\Omega \left({(\log n/\log \log n)}^{1/(d-k)}\right)$ of them. As a corollary we deduce that every d-polytope with n vertices has a k-shadow with $\Omega \left({(\log n/\log \log n)}^{1/(d-k)}\right)$ vertices.

In 2002 Gardner and Gronchi obtained a discrete analogue of the Brunn-Minkowski inequality. They proved that for finite subsets $A,B\subset R^n$ with $\dim B=n$, the inequality $|A+B|\ge |D_{\left|A\right|}^B+D_{\left|B\right|}^B|$ holds, where $D_{\left|A\right|}^B,D_{\left|B\right|}^B$ are particular subsets of the integer lattice, called B-initial segments. The aim of this paper is to provide a method in order to compute $|D_{\left|A\right|}^B+D_{\left|B\right|}^B|$ and so, to implement this inequality.

A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A graph G belongs to class W_n if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W₁ is the family of all well-covered graphs. Staples showed a number of ways to build graphs in W_n, using graphs from W_n or W_n+1. In this paper, we construct some more infinite subfamilies of the class W₂ by means of corona, join, and rooted product of graphs.