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.

The total distance or Wiener index $W\left(G\right)$ of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991, Šoltés posed the problem of finding all graphs G such that the equality $W\left(G\right)=W(G-v)$ holds for all their vertices v. Up to now, the only known graph with this property is the cycle C₁₁. Our main object of study is a relaxed version of this problem: Find graphs for which total distance does not change when a particular vertex is removed. We show that there are infinitely many graphs that satisfy this property. This gives hope that Šoltes's problem may have also some solutions distinct from C₁₁.

Given two sets of points A and B in a normed plane, we prove that there are two linearly separable sets $A^{\prime }$ and $B^{\prime }$ such that $\mathrm{diam}\left(A^{\prime }\right)\le \mathrm{diam}\left(A\right),\mathrm{diam}\left(B^{\prime }\right)\le \mathrm{diam}\left(B\right)$, and $A^{\prime }\cup B^{\prime }=A\cup B$. As a result, some Euclidean clustering algorithms are adapted to normed planes, for instance, those that minimize the maximum, the sum, or the sum of squares of the diameters (or the radii) of k clusters. Some specific solutions are presented for $k=2$ and $k=3$ that minimize the diameter of the clusters. The 2-clustering problem when two different bounds are imposed to the diameters is also studied.