European Journal of Combinatorics最新文献

The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $S\subseteq V\left(G\right)$ so that for any two vertices $x,y\in S$ there is a shortest $x,y$-path $P$ so that all internal vertices of $P$ are not in $S$. This is also said as $x,y$ are visible with respect to $S$, or $S$-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside $S$. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters.

The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankiewicz’s problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Turán problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size.

We study the minimum number of distinct distances between point sets on two curves in $R^3$. Assume that one curve contains $m$ points and the other $n$ points. Our main results:

(a) When the curves are conic sections, we characterize all cases where the number of distances is $O(m+n)$. This includes new constructions for points on two parabolas, two ellipses, and one ellipse and one hyperbola. In all other cases, the number of distances is $\Omega (min\{m^{2/3}{n}^{2/3},m^2,n^2\})$.

(b) When the curves are not necessarily algebraic but smooth and contained in perpendicular planes, we characterize all cases where the number of distances is $O(m+n)$. This includes a surprising new construction of non-algebraic curves that involve logarithms. In all other cases, the number of distances is $\Omega (min\{m^{2/3}{n}^{2/3},m^2,n^2\})$.

Let $X$ be an $n$-element set. A set-pair system $P={\left\{(A_i,B_i)\right\}}_{1\le i\le m}$ is a collection of pairs of disjoint subsets of $X$. It is called skew Bollobás system if $A_i\cap B_j\ne 0̸$ for all $1\le i<j\le m$. The best possible inequality $\sum _{i=1}^m\frac{1}{\left(\frac{\left|A_i\right|+\left|B_i\right|}{\left|A_i\right|}\right)}\le n+1.$ is established along with some more results of similar flavor.

In this paper, we explore the combinatorics behind an identity recorded in Ramanujan’s lost notebook. We present an interesting result, which not only generalizes two theorems of Bressoud, but also implies a bivariate form of Ramanujan’s original identity.

We obtain estimates for the number $p_d\left(n\right)$ of $(d-1)$-dimensional integer partitions of a number $n$. It is known that the two-sided inequality $C_1\left(d\right)n^{1-1/d}<log{p}_d\left(n\right)<C_2\left(d\right)n^{1-1/d}$ is always true and that $C_1\left(d\right)>1$ whenever $logn>3d$. However, establishing the $“$right$”$ dependence of $C_2$ on $d$ remained an open problem. We show that if $d$ is sufficiently small with respect to $n$, then $C_2$ does not depend on $d$, which means that $log{p}_d\left(n\right)$ is up to an absolute constant equal to $n^{1-1/d}$. Besides, we provide estimates of $p_d\left(n\right)$ for different ranges of $d$ in terms of $n$, which give the asymptotics of $log{p}_d\left(n\right)$ in each case.

Given graphs $G$ and $H$, we say $G\stackrel{r}{\to }H$ if every $r$-colouring of the edges of $G$ contains a monochromatic copy of $H$. Let $H\left[t\right]$ denote the $t$-blowup of $H$. The blowup Ramsey number $B(G\stackrel{r}{\to }H;t)$ is the minimum $n$ such that $G\left[n\right]\stackrel{r}{\to }H\left[t\right]$. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given $G$, $H$ and $r$ such that $G\stackrel{r}{\to }H$, there exist constants $a=a(G,H,r)$ and $b=b(H,r)$ such that for all $t\in N$, $B(G\stackrel{r}{\to }H;t)\le a{b}^t$. They conjectured that there exist some graphs $H$ for which the constant $a$ depending on $G$ is necessary. We prove this conjecture by showing that the statement is true in the case of $H$ being 3-chromatically connected, which in particular includes triangles. On the other hand, perhaps surprisingly, we show that for forests $F$, there exists an upper bound for $B(G\stackrel{r}{\to }F;t)$ which is independent of $G$.

Second, we show that for any